Jonathan Miller's dazzling career has been built on paradox. As a younger man, he was expected to become a brilliant scientist and his schoolfriend Oliver Sacks claimed he was "much more gifted than I was". Instead, he pursued a career in "the footling flibbertigibbet world of theatre", most notably as part of the Beyond the Fringe quartet, before quitting comedy to direct opera and television documentaries.

In Kate Bassett's absorbing, in-depth and erudite biography of Miller, the first major one published, he is shown to be someone who thrives on confounding expectations, even if the impression given is unflattering. Despite having been vice-president of the Campaign For Homosexual Equality, for instance, he isn't shy of referring to "disgusting old opera queens", a remark that would lead his erstwhile Fringe colleague Alan Bennett to call him "the Enoch Powell of the sexual world".

Bassett's book is not an authorised biography, but draws on conversations with Miller, his friends, family and enemies (of whom there are more than a few) to produce an appealingly multifaceted figure of a man whose polymathic genius has never been in doubt, but whose refusal to toe the British line of self-deprecation (unlike Bennett) has seen him become something of a prophet without honour at home.

His friend John Fortune's cited remark that "if he'd been born French, there would be streets named after him" is distinctly double-edged. The impression given throughout is that Miller, still a vital and energetic presence as he nears 80, has a high impression of his intellectual standing  "I'm the only person worth competing with," he once said  and little time for those who disagree. Bassett, who clearly likes and respects her subject but isn't overawed by him, sensibly marshals the arguments on both sides and leaves it up to the reader how posterity will regard Miller.

In Jonathan Miller's brilliant, bubblegum-pink 2010 reimagining of L'elisir d'amore for English National Opera, there was a moment that caused bel canto purists to clutch for their heart pills. It came  from memory  early on. Adina, the heroine, was reading aloud from the story of Tristan when the orchestra abruptly abandoned Donizetti's delicate oom-pah-pah and lobbed in 19th-century music's most notorious hand grenade: Wagner's Tristan chord. It was a joke, but a serious one: half schoolboy prank, half learned footnote. If you could enjoy a gag this highbrow, it was also shamelessly funny.

Miller adores the element of surprise, and not only on stage. Impressionist and satirist; raconteur and chatshow darling; director and producer of opera, theatre and TV; quondam boss of the Old Vic; presenter, author and scriptwriter; lecturer, curator, artist  a dizzying life, and that's putting to one side his training as a doctor, which has impelled projects as varied as a BBC series on the history of medicine, The Body in Question (1978), and medically exact productions of Così fan tutte. A zealous non-believer, he has produced a shattering version of the Matthew Passion; riffed casually on his own Jewishness ("not really a Jew  Jew-ish") yet also staged one of the most thoughtful examinations of that faith in living theatrical memory  his 1970 Merchant of Venice with Laurence Olivier, performed as Miller's own father, a devout Jew, lay dying. As Kate Bassett's thorough, well-upholstered new biography makes plain, Miller contains multitudes. More than perhaps even he is aware of.

He was born into, if not quite greatness, then the heavy expectation that he would achieve it. His father Emanuel was a well-regarded child psychiatrist; his mother, Betty, a novelist and biographer. Miller fils, somehow appropriately, didn't wait until birth to make his literary debut  making a cameo, Bassett suggests, as an unborn child in one of his mother's stories. The cultured milieu of his north London childhood is hinted at by the fact that Stevie Smith once wrote a satirical piece about his family (Bassett calls Smith's accompanying poem "vile").

It was to science that the young Miller was drawn: first at St Paul's school  pals included Oliver Sacks and book-dealer Eric Korn  then at Cambridge. But almost as soon as he began his medical studies, the "cocaine-like" addiction of performing was working away: skits at school and university, then, after graduation, the astonishing success of Beyond the Fringe  a quartet, it's pleasing to be reminded, that was assembled by a young theatre producer, as coldly manufactured as any boyband. Miller's later claims that his career has been largely happenstance are cast into relief by a letter he wrote at the age of 18 to the BBC requesting a TV audition, explaining that he and a friend "specialise in parodies of Radio, films and theatre and also 'REAL LIFE'" (the request was turned down).

Bassett's thesis is encapsulated in her title, and she spends much time examining, then re-examining, Miller's bursts of spirited indecision about what to do with his talents  no sooner promising to knuckle down to medicine before scarpering towards the footlights; no sooner triumphing in theatre before announcing a penitential progress back towards science. Bassett makes the fine point that doctoring and directing aren't perhaps so different, over and above simply requiring theatres. Ibsen, Chekhov, Goldsmith, Strindberg and Schiller all received medical training, and you could argue that both disciplines involve careful performance rituals and a degree of flamboyant mystification. Both also call for deep involvement in human problems; and practitioners of both kinds sometimes fancy they have the god-like ability to transcend them (even if, as Miller discovered, the West End proved more tempting than ward rounds at UCH).

Although Miller seems to have realised he is ill-equipped for the slog of medical research (something it apparently took a £88,000 grant in the 1980s at Sussex University to demonstrate), his dividedness persists, as evidenced by numerous declarations that his directing career is over  he has enjoyed more curtain calls than the most prima of prima donnas. It is typical  also somehow heartbreaking  that, not content with being a hobbying sculptor, Miller rather fancies installing what is described as a "large-scale work" outside Waterloo station. This isn't common-or-garden hypercompetitiveness; it seems to be a kind of mania, on a literally architectural scale.

This, alas, is the frustration of In Two Minds, which gives the recurring impression of leading its subject towards the psychologist's couch without ever eliciting sustained or meaningful analysis. Miller's relationship with his sister Sarah receives minimal treatment, his children feature largely as a support act; give or take the odd hint, the realities of his 56-year marriage are placed discreetly behind a curtain (his wife Rachel is a GP, and some of the professional challenges she's had to face sound a good deal tougher than her husband's). We are halfway through the book before we discover that Miller has suffered from depression: the discussion of how that illness has manifested, or coloured his work, lasts less than a page.

Being in two minds about one's career is one thing; the more interesting question, surely, is how the schisms in Miller's own mind fit  or don't fit  together. Which of his personalities does one believe? The kindly friend who helps a colleague move house, or the man who cuts former acquaintances dead for daring to contradict him? The workaholic professional, or the one who abandons rehearsals when they aren't going his way? Vice-president of the Campaign for Homosexual Equality, or the man who cracks jokes about "opera queens"? The nuclear-powered controversialist? The penseur and philosophe? The licensed fool?

Bassett's acknowledgments say that Miller himself has not read or edited any of the manuscript; he hardly needed to, such is the flood of his talk that courses through it. Hefty verbatim quotes form its substance, and even ostensibly independent opinions (that telling "vile"; leitmotifs on his parents' lack of affection, an odd tic to do with never having earned enough) have the tang of indirect speech. Sometimes this is innocuous enough  and often it's entertaining, Miller's tongue being famously sharp  but despite Bassett's attempts at balance, the suspicion grows that we are being drip-fed the subject's own insights into himself. Richard Holmes once compared the biographer's task to snooping on a dinner party where the guests are permanently out of earshot; Bassett brings us right to the table, but seems anxious about offending the host.

Bassett is herself a fine, fierce theatre critic  her write-ups of Miller's productions have verve and perceptive grace  so it is a shame that, when it comes to the great man, she too often abandons her critical distance, notably in a final section that marshals witnesses for the defence (John Fortune: "If he'd been born French, there would be streets named after him"; the historian Michael Wood: "the most scintillating and inspiring teacher  bollocks to those who think otherwise"). Alan Bennett, a long-term neighbour as well as family friend, is more wary: "We've always gone on the assumption that the less I said about him and the less he said about me the better."

In the end, perhaps one of the most revealing character sketches here is a short section on Private Eye, which for some years ran a column called "The Life of Dr Jonathan", casting Miller ("the Great Doctor") as a latterday Johnson, dilating at length to "importunate savants" on "topics too awful and profound to bear thinking of". In the backhanded way of British satire, it's as close as Lord Gnome's organ has ever come to unadorned compliment. Miller, naturally, cannot see the funny side.