Carter, an American mountaineer, scaled Everest as part of a project to determine the effect of high altitude on climbers' cognitive abilities. He wasn't feeling too clever on the way up, but on the way down his condition rapidly deteriorated.
He takes up the story. "I had a common cold, but at altitude the phlegm in my throat was coagulating, and I was choking to death. Ed Viesturs [his companion] had to do the Heimlich Manoeuvre on me for six hours.
"Then I went into respiratory arrest, and Ed, who is a vet, was preparing to do a tracheotomy on me. I was ready to go and die, when suddenly I thought, `Hell, I'm not going to.' I put my hand down my throat and ripped out a brownie-sized piece of phlegm. It was the best thing that has ever happened to me because it made me realise that getting to the top of Everest was no big deal. I'm the luckiest son of a bitch alive, and I'm a better person for it."
This shockingly compelling incident forms the climax to "Thin Air", a gripping contribution to C4's Equinox strand. The aim of the documentary, by David Breashears and Liesl Clark, was, according to Carter, "like an old anti-drugs commercial that we had in the US. It showed an egg frying in a pan, and said `this is your brain on drugs'. Everest does the same damn thing; it's probably not a real good thing for your brain."
Above 17,000ft, the air contains 65 per cent less oxygen than it does at sea level. The body rapidly starts to decline from the combined effect of muscle wastage, sleep loss and weight loss. From 19,500ft onwards, Carter is visibly flagging. "It's slow, you're winded, dehydrated, you're losing your voice, coughing. But the views make it worthwhile."
When you climb above 26,000ft, you are entering what is known as the "death zone", where humans cannot survive for long. Such is the effort required even by the act of breathing that the last 300ft of the mountain takes two hours to climb. The 29,028ft summit, which pushes out into the jet stream, is littered with corpses of climbers less fortunate than Carter. No-one has enough spare energy to carry them down. For Breashears, on his fourth trip to the top, seeing the bodies "only makes me question my sanity".
In the film, Carter, Breashears and Viesturs are depicted struggling at altitude to repeat simple sentences. In a "true or false" test, Carter is shown saying it is true that "lion is a military title".
But, beyond the scientific trials, the trip had valuable life lessons. "I was a member of the best team in the world," Carter says. "I was surrounded by the Michael Jordans of high-altitude climbing. It made me realise that whatever you do in life, it's important to be able to work as a team. I climb because I enjoy working with a team that's focused on the same goal - you don't always get that in business. A true climbing team is only as strong as its weakest member."
Perhaps because of his brush with death, Carter also realised there are some "spiritual aspects" to it.
"Everest is a living thing, and it picks and chooses who gets to the top and who survives. You have to believe in the mountain and pay attention to its rituals. Every hour I think about that climb - some titbit will come into my head. I'm an aggressive, Type-A person, and it makes me slow down and smell the roses. I left part of myself up Everest. When I die, my spirit will float right over that mountain."
Carter has promised his family that he will never attempt to reach the summit of Everest again, and Breashears has vowed the same - "I need lots of people to prevent me from changing my mind".
But Carter has no time for the doom-mongers who assert that you should never try it in the first place. "People say, `you must have a death wish', but I laugh and reply that I do it because I love life. I'll never just sit around on my ass. Next week I'm going climbing in Russia."
Nor does he agree with people who claim that high-altitude climbing is a selfish pursuit. "I hear that all the time, but life would be pretty damn boring without adventure. Nothing would be accomplished without risk. Look at Lindbergh, or the astronauts. Explorers have taught us so much."
So what has Carter learnt from the experience? "Being on Everest was the best time of my life, even though I was in mortal danger. Pushing yourself to the limit is a neat feeling. It shows me that anyone can do it.
"There are lots of Everests out there for people. My Everest just happened to be Everest."
`Equinox: Thin Air' is on Tue at 9pm on C4