But even he, a master of the unexpected, would have blinked at some of the evidence on the investigators' file. Crows, owls and hawks dropping from their perches in the Bronx zoo. Horses being 'euthanised' across the United States. A mad rhinoceros breaking its neck. Soldiers paralysed by their soup.
He was 32 when life finally caught up with him and his enemies could enjoy the spectacle of his exquisitely lingering death - a death that, in the opinion of a leading forensic psychiatrist, had been made inevitable by the loss of his long-term gay lover.
According to an inscribed clay tablet in the British Museum, Alexander the Great breathed his last on June 11, 323BC. The event is laconically recorded - "there were clouds and the king died" - though there can have been no doubt about its world-changing enormity. Nobody at the time knew what had finished the tyrant off, but - given that his last conscious act had been to drink himself stupid - there was more than a suspicion that strong Macedonian wine had done its worst.
But it leaves open the crucial question: why did he take 12 days to die? Modern pathology suggests very few causes of death that fit the pattern, and none of them involves red wine. Some of them, however, do involve poison. If you are an experienced 21st-century policeman, this is exactly the kind of thing that makes your antennae twitch. You have a death. You have suspicious circumstances. You have a lot of people with good reason to dance on the victim's grave. You have a case.
Not a hopeless one, either, for although there is no body to examine, and no crime scene, there is plenty of evidence that has never faced the scrutiny of a professional investigator wise in the ways of unnatural death.
John Grieve, CBE, QPM, BA(Hons), M Phil, is wise in other ways too. When he retired in May 2002 he was deputy assistant commissioner of the Metropolitan Police. What he left behind him was not so much a track record as a comet tail. A reputation for the unorthodox, plus cerebral tastes in literature and art, his rumpled appearance and lugubrious manner, brought the inevitable comparisons to Inspector Morse. He groans at the memory. "When journalists want me to be Morse," he says, "I drive an old Jaguar, drink warm beer and like opera. The fact is, I don't drive, I like rock'n'roll and I don't touch beer because I'm a diabetic."
There are more important differences too: his career in the Met took him to places that Morse could only have had nightmares about. As head of anti-terrorism he cracked two of the IRA's top operational units and effectively put an end to the mainland bombing campaign. As head of the Yard's Racial and Violent Crime Task Force he had the unenviable but vital task of formulating the Met's response to the Stephen Lawrence inquiry. Further back, he had pitched himself against the criminal gangs of London's East End, and knocked off murderers and drug dealers with the efficiency of a crime-seeking missile.
The commissioner of the Metropolitan Police, Sir John Stevens, did not risk controversy when he described him as "one of the best, if not the best, detectives ever to have worked at Scotland Yard".
And now here he is, at the invitation of a film company, leading a worldwide investigation into a fatality that occurred 323 years before the birth of Christ. He treats the job the only way he knows: seriously. "It is an opportunity," he says gravely, "to stop and think about the philosophy of investigation."
To hear him speak, you might believe he never thinks about anything else. Sitting in the west London conference room of the film company he is working with, Atlantic Productions, he is in no mood to be hurried. His intention, he says, is to assemble enough evidence to put before a jury - and by "evidence" he means exactly that. He'll not be satisfied with conjecture, no matter how eminent the historian putting it forward, and he is not going to cut corners. "I don't care how obvious it may look to other people," he says. "I'm maintaining an open mind." It's not that he dismisses historical orthodoxy: rather that he regards it as just another tool in the box. At dictation speed he recites the list of specialisms he will bring into play: "Forensic science, pathology, psychiatry, logic, medicine, botany, ethics, epidemiology, history, toxicology, archeology..."
Like any policeman at a briefing, he refreshes himself from his notebooks. Here is a surprise, though. Instead of the expected scrawl, there are beautiful coloured illustrations and carefully lettered diagrams in a vivid spectrum of inks. He seems genuinely surprised that I don't recognise them for what they are: mind maps. These, he says, are one of the many things in his life that (a favourite expression) have "bent his head" - a graphic and, hence, more memorable way of recording information.
On the page, he prints a main subject heading in bold capitals. Lines radiating from this are labelled with the most important subheadings, with smaller branches and twigs to represent lesser subdivisions, each one crisply identified by a single word, concise phrase or graphic symbol. Different colours help both to clarify the map and to make it easier to remember. Being a talented amateur painter, Grieve adds impact with coloured miniatures that catapult the practice of criminal investigation across the line from science to art. Small wonder that "creativity" occurs so often in his conversation.
