1 Caption: ==Mythbusters== Quantum Immortality
2 Jamie: The many-worlds interpretation of quantum mechanics says that every time different outcomes of an event are possible, the universe bifurcates.
3 Jamie: This creates one universe in which one outcome occurred and one in which the opposite outcome occurred.
4 Jamie: The upshot of this is that whenever you might die, you actually die in only one universe, but live on in another, parallel universe.
5 Adam: So if you follow the right chain of universes, you can never die?
5 Jamie: Right!
6 Adam: So... while living forever, we can also die multiple times?
6 Jamie: Yes!
7 {beat}
8 Adam: Is there any universe in which this actually makes sense?
8 Jamie: Only a few of them.

No, really, this is a serious argument used to discuss the philosophical implications of quantum mechanics.

In its simplest form, the quantum immortality argument concludes that:

IF the many-worlds interpetation of quantum mechanics is true, THEN conscious beings are immortal when the ensemble of multiple universes is considered.

EDIT: Okay, I got a bit confused there - stop sending e-mails. :-) The upshot of this is that - from your own perspective - you either die at some point, or continue living. If you die, you cease to experience things, but in a world in which you don't die, you naturally continue living and experiencing the universe around you. Which means that if you follow your own stream of consciousness, there exists a universe in which you never die. In other words, we are all immortal (in some universe or other)! This is a difficult point to test, because we all inhabit "this" universe, whatever that means. But the conclusion is that either, (a) somewhere amongst the many worlds created by quantum mechanics you will live forever, or (b) the many-worlds interpretation of quantum mechanics is a load of bollocks.

Remember that last point...

[EDIT: I'm sure if I've got it wrong again, you'll let me know. The really important thing here is that either (a) something completely ridiculous, or (b) the many-worlds interpretation of quantum mechanics is a load of bollocks. Feel free to make up your own completely ridiculous thing. :-) For instance, try the following: ]

There are also various arguments that the quantum immortality thought experiment is flawed for other reasons - you can check some on the Wikipedia page.