When Stewart finally departs, he will go with his integrity intact. If he had just wanted to do "something shinier somewhere else", he could have joined Meet the Press. NBC reportedly offered the New Yorker an open cheque to make the crossover into legitimate news programming, but Stewart resisted.
"That," he says, "was just one of those things where they go, 'We really like this thing that you do. So here's what we'd like you to do: come not do it for us. Do something completely different.'"
Straight reporting wouldn't have suited him, anyway; he needs to leaven the seriousness with laughter. "It is wiring, to some extent," he explains. "I think it's just how I cope: repression and humour."
Ben Karlin, Stewart's former production partner on The Daily Show, once noted that Stewart (born Jonathan Stuart Leibowitz) was bullied as "a rare Jewish kid" growing up in a New Jersey neighbourhood. Stewart, though, dismisses any suggestion that he might have developed his comedic outlook as a response. The abuse, he insists, really wasn't that bad or unique.
"This was not a Louis Malle movie. It was no more than what the Italian kids in my school faced for being Italian. Or the Irish kids. And because of the way my brain is wired, I think I tended to deflect it with humour. Other kids, genetically superior to me, would use fisticuffs."
Rosewater, based on the memoir Then They Came for Me: A Family's Story of Love, Captivity and Survival, by Maziar Bahari and Aimee Molloy, likewise blends light and dark in its retelling of how the Iranian-Canadian journalist Bahari spent 118 days in solitary confinement in Tehran's notorious Evin prison, undergoing torture and intensive interrogation.
Bahari never lost sight of the absurdity of his situation, making his story perfect territory for Stewart. Even more to the point, Bahari's arrest was partly due to his connection with The Daily Show.
Stewart's team had been trying to get into Iran ever since George W. Bush designated it the "axis of evil". "At the show," he says, "that was a relatively irresistible, two-dimensional moniker that made us think, 'We must go to this place to see evil with our own eyes." They tried for years, without getting access.
Then, in 2009, Iran liberalised its entrance visa requirements for journalists in the run up to the presidential election pitting the moderate Mir-Hossein Mousavi against hard-line incumbent Mahmoud Ahmadinejad. "So we snuck in on that," says Stewart, "having no real intention of covering it."
Bahari was in Tehran reporting for Newsweek, and The Daily Show's Jason Jones hooked up with him for a sketch in a café, in which the comedian, wearing a kaffiyeh and sunglasses, pretended to be an American spy. Later, when Bahari was arrested, this was used as evidence against him.
The absurdity is summed up in the film when Bahari (Gael Garcia Bernal) explains to his interrogator that Jones is a comic. "Why does this man claim to be a spy if he is not a spy," asks the inquisitor. "Why would a spy have a television show," responds Bahari.
No one imagined that working with Bahari would put him in danger, because the café scenario was "so truly ludicrous", says Stewart. He often gets asked whether he felt guilty about what happened, and he admits that it is something that he has often thought about. Nevertheless, his conscience is completely clear.
"The problem of responsibility is that what we did was nothing. What Maziar did was nothing. He wasn't actually a spy. Our interview with him was not part of a conspiracy," he says.
"When a country weaponises the banal or the innocuous or idiocy, you try and think to yourself, 'What could we have done differently?' And the answer is, nothing. Because whatever pretence they were using was false."
Bahari's real ''crime'' was that he filmed and then passed on footage of the killing of a young protester by security forces during the demonstrations that erupted in the wake of Ahmadinejad's shock victory.
"So what he really got in trouble for was witnessing atrocity," says Stewart.
"And I guess what we could have done differently is not have anyone witness atrocity."
Ultimately, bearing witness is what Rosewater is about. Regimes can throw people in prison, torture them and force them - as they did with Bahari - to confess on television.
But, as the film shows at the end, their efforts to control people and the flow of information can be undermined in an instant by a child with a camera phone.
This is the perfect punch-line for a film made by a writer-director once dubbed "the most trusted man in America". Whether Noah will be able to fill his shoes remains to be seen. But, as Stewart wasn't just a presenter but a writer, too, The Daily Show is unlikely to be the same after he leaves later in the year. His plans for the future remain vague (he has "projects" on the boil and would like to make more films), but we can be sure that whatever he does, it is bound to raise a smile. After all, Stewart can't help himself: "I just think I'm generally wired to pratfall and pun."
