Wikipedia: Quantum immortality
In the Schrödinger's cat experiment, what would happen from the cat's point of view? Would the cat's consciousness be split in two, with one consciousness living and the other vanishing into the sweet hereafter? Or would the cat, having the special status of being an observer, perceive itself as following the more fortunate fork in the quantum road and thus avoid death?
To quote the Wikipedia page:
The idea comes from a variant of the quantum suicide thought experiment. Suppose a physicist standing beside a nuclear bomb tries to detonate it. In almost all parallel universes, the nuclear explosion will vaporize the physicist. However, there is a small set of alternate universes in which the physicist somehow survives. The idea behind quantum immortality is that the physicist is only alive in, and thus able to experience, one of the universes in which a miraculous survival occurs, even though these universes form a small subset of the possible universes. In this way, the physicist would appear, from a personal point of view, to be living forever.
We should all hope to Ghawd that quantum immortality is a mere figment of the physicists' imaginations.
This wouldn't be immortality for one and for all. Only one person in the universe would benefit -- namely, you!
Your consciousness would never cease, but all those around you (whose consciousness you don't have the privilege of sharing) would be subject to the normal limits of biology. But from their point of view, they would each be immortal, and all those around them, including you, wouldn't.
It would be a perverse form of immortality. You would live to see your family and friends grow old and die. You would live to see civilization rise, and fall. You would live to see the end of the human race. You would live until the Sun went supernova and the Earth was vaporised into a cloud of ash. And still you'd live.
Maybe "live" isn't the right word. Technically, your consciousness would continue. That says nothing about the body. Since your consciousness is seated in the brain, there's no need for the rest of your body to exist.
A trillion years from now, only your brain would remain, quivering, spinning through the void, conscious (alive?) only because of an infinitely improbably sequence of quantum events that lets your synapses keep firing.
Thanks to the many worlds interpretation, you wouldn't be alone (in a conceptual sense at least). The same fate would befall every sentient being who has ever existed, and will ever exist, in the cosmos. You would have the cold comfort of knowing that, in an uncountable infinity of eternally disconnected universes, others are experiencing exactly what you are.
And I'm sure they'd all be thinking the same thing: "CURSE YOU, QUANTUM MECHANICS? WHY WILL YOU NOT LET ME DIE? WHYYY?"
For in the Church of Quantum Immortality, there is no Heaven -- only Hell.