According to those close to her, Beryl Bainbridge was striving to finish her last novel shortly before she died last July. Not only does it seem an entirely natural impulse in a lifelong writer confronting the final line, but The Girl in the Polka Dot Dress adapts one of story-telling's most established narratives, the quest. How, then, to be satisfied with creating a search without a discovery, a journey without an arrival?

Except that Bainbridge was never quite that neat a writer; elliptical, mysterious and not too hung up on the indispensability of closure, her novels quite frequently seemed to lack an easily decipherable resolution, and be all the more powerful for it. This, her 18th, does indeed seem to have been interrupted by her death; her long-term friend and editor, Brendan King, prepared the text for publication from her working manuscript, "taking into account suggestions that Beryl made at the end of her life". But despite the novel's climax tending to be febrile and incomplete - not only bringing to an end the journey of its two central characters but also encompassing the assassination of Robert Kennedy - what remains is a characteristically dark and mischievous slice of Beryl at her best.

We begin in confusion. Rose, a dental receptionist who has been side-stepping the advances of dubious men ever since she arrived in London at 16, a refugee from an oppressively unhappy childhood, is now on the move again. Almost 30 but strangely childlike, her destination is America but may as well be the moon, so adrift does she seem; and her host, the daffodil-bearded Washington Harold, immediately strikes one as inadequate as a protector. But why are Rose and Harold, who barely know one another, poised to journey from Baltimore to California in a second-hand camper van, albeit one with running water, and an Abraham Lincoln clock? And why are they so intent on tracking down the trilby-hatted but otherwise almost featureless Fred Wheeler?

Clues come quicker than answers. Rose regards Wheeler as her saviour, the catalyst that allowed her to escape the parents who blighted her childhood and were responsible for the adoption of the child she bore when under age; he appears to have had no direct agency, but rather imbued her with the liberating apprehension that "suffering was the direct and immediate object of life", and that the world is a penal colony where a price must be paid for existence. Harold's view of Wheeler is rather different: his metamorphosis from implausibly glamorous and powerful friend to wife-stealer means Harold's mission is one of revenge rather than reunion, a fact he is at pains, throughout, to conceal from Rose.

Wheeler, however, is not all that divides them. Disappointed by one another almost from the off, they make curious travelling companions, rumbling from Baltimore towards Los Angeles where, it is rumoured, Wheeler has become part of Kennedy's entourage ahead of the California primary  in a state of mutual incomprehension that often shades into disdain. Of Rose's gnomic pronouncements, mystical flights of fancy and incuriosity towards her surroundings, Harold "told himself that if he wanted to avoid slapping her he must bear in mind that he was dealing with a retard".

For her part, Rose wishes merely to keep on the road, homing in on the almost existentially elusive Wheeler; Harold's expectations of her - he envisages her tossing salad while he points out the wonders of the night sky and then, we imagine, the rest - are of little concern to her. And we don't entirely blame her. "Trust Harold," remarks one of his bohemian friends when Rose reveals she has been taken on a less than thrilling tour of Washington. "Always the man for exciting information."

But if Rose finds America moving wallpaper, then Bainbridge does not. She conjures a country in the wake of the assassination of Martin Luther King, where fires burn, workmen board up shops and red paint is hurled at synagogue doors. On the road, she describes "a confusion of flyovers, underpasses, intersections, junctions, toll gates. Yield, the signs instructed in bright yellow. Sometimes there were fields full of cows, once a river, brown and swollen, once a town with a railway track running down the middle of its street. On either side, bursting back from the highway, the trees tossed rainwater." On a postcard to her friends back in London, Rose writes simply "weather lovely".

From campsite to campsite and diner to diner, through groups of Harold's eccentric friends, Harold and Rose encounter a series of truncated obstacles, giving a lift to a monsignor who's broken down on the way to bury a soldier flown back from Saigon, falling foul of a Theosophist and being held up in a bank robbery. But the ups and downs of their off-kilter road trip are not Bainbridge's real focus, comically and deftly sketched though they are. Her real talent is to show minds glancing off one another, all of them hedged around with fear and desire and the shadows of the past. For Rose and Harold, each stuck, in their different ways, in a history that seemed to run away from them, it is a ridiculous irony that they should be attempting to journey into the future. It was never going to work.

It's no accident that when Rose and Harold reach Los Angeles they intersect with an episode from American history not only so painfully fateful but also so bewildering. The brief appearance of Sirhan Sirhan - Bobby Kennedy's assassin, whose actions have been the subject of theories that include hypnotism and mind control - pulls us towards dizzying thoughts of individual responsibility and the effects of charisma and personal magnetism. The Girl in the Polka Dot Dress may not have every final i dotted and t crossed but, as most of Bainbridge's oeuvre did, it leaves its readers with more to think about than one might imagine possible for such a slender tale. It is a fitting finale and a poignant farewell to a career defiantly and uncontestably sui generis.

When Beryl Bainbridge was on what proved to be her deathbed, entangled in tubes, she was insistent that I (and I'm sure others) get the doctors to give her 30 days in which to finish her novel.

Beryl moved uniquely on the cusp between a finely self-mocking wit and steely seriousness. Sometimes it was difficult to know quite what was going on. And was her protestation something of a performance to help us, her visitors, through the pain of watching her inexorable exit?

The Girl in the Polka Dot Dress shows that she was not only serious, she was desperate. She was obsessed with completing the novel and the novel aches for completion. Her long-time friend and editor, Brendan King, has put together the book, we are told, "from her working manuscript, taking into account suggestions Beryl made at the end of her life. No additional material has been included." When I finished the book for the second time I wanted to write to him and say: are you sure there is no additional material? Just a few clues? Any clue will do.

What we have is an unfinished novel that blazes with Beryl's unique talent. The story is teased out stealthily and you have to be on the lookout. She wastes not a breath on unnecessary explanation. In brief, two people, Rose from England and Harold from America, drive across the latter's continent looking out for a Doctor Wheeler. Rose is about 30, but she met Doctor Wheeler, an American, in Liverpool when she was a young girl and one way or another they kept in touch. Wheeler has become a tutelary figure and almost mythic in her memory. She has got herself invited to America by Harold, a rather shadowy character who is using her to get to Doctor Wheeler in order, we learn eventually, to take revenge on him. Wheeler has, we understand, had an affair with Harold's wife, which was followed by her suicide.

They drive across America in pursuit of Wheeler, who is always a city or two ahead of them. There is something of the road movie about the novel as we track through a dysfunctional country in an apocalyptic phase. Robert Kennedy is about to run for president. Assassinations are the nature of the times.

Harold is an intellectual and we keep meeting friends of his who want to talk politics. Rose is delphically self-absorbed and to Harold's horror seems to be banal, almost retarded

Rose is on the way to being one of Beryl Bainbridge's greatest creations. She comes out of a violent background  violent in words and manners, but violent also in the way in which her life has been decided for her. A love affair at 15 (the only time she was in love) with a boy her own age resulted in a child, ripped away for adoption at birth. Her father's weird anger drove her out of the house repeatedly. There was at least one violent rape attempt. Doctor Wheeler is the only beacon she has.

What is wonderful about her is the way that she is given an internal life through sentences that are brilliantly elliptical and often very funny. When Harold offers her a glass of wine she shakes her head: "She wasn't into wine; in her opinion it took far too long to make one cheerful." "The here and now meant little to her; it was what made her so unusual." She reflects that "judging from the state of the toilet bowl, Americans didn't know about Vim". And then she will come out with something that shakes Harold: "It never does any good," she said, "to dwell on things that can't be changed. That way madness lies." She is very reluctant to wash, either herself or her clothes: "Too much cleaning makes us susceptible to germs," she says authoritatively.

Rose regularly converses with her past and mumbles conversations to her dead father. The intercutting between past and present works like a dream. On occasions when Harold and his friends talk politics, she takes part in the conversation but follows her own independent line. For instance, they are speaking about Senator Joseph McCarthy and Rose pops up about "that song about a park in a rainstorm". She goes on to sing, "Macarthur's Park is singing in the rain ..." The others ignore her and continue the higher gossip while she continues on her own sweet way.

It becomes clear that Harold is out to kill Wheeler. Rose, the waif, defeats any attempt he can muster to engage her, although typically when he does once have sex with her for a few seconds she is quite pleased because she feels less bad about the money he is spending on her, taking her across America in his camper van. A debt has been paid.

Brief encounters mark their journey. At one place they are cooked a lunch. The lady of the house, Philopsona, offers them chicken: "'The birds,' she trumpeted, were her pride and joy, each one with a name and fondled from birth. She never allowed anyone but herself to wring their necks. 'It wouldn't be right,' she assured Rose. 'They need someone they can fucking trust.'" Not long after that the novel ends. We are then presented with a (real) report from the Los Angeles Times, 6 June 1968, which speaks of a "girl in a white dress with polka dots who ran from the hotel where Senator Robert F Kennedy was shot and said 'we shot him'." When we left Rose she was wearing a polka dot dress and had somehow got into the Ambassador Hotel. When we left Harold he had shaved off his distinctive bushy beard. We knew he had a gun and he had severed his link with Rose.

Yet the newspaper report poses questions. If "Rose" did say "we shot him", who does the "we" refer to? And was the phrase "we shot him" said in triumph or panic or just another example of Rose hitting the nail on the head and moving on?

The problem is that between the novel and the LA Times report you really want to see the join. This does not stop The Girl in the Polka Dot Dress from being a superb and memorable work of fiction, even though it lacks the final completeness Beryl longed to give it.