Quantum immortality, while obviously speculative, is a legitimate scientific notion, at least to the degree that all questions about the interpretation of quantum mechanics are. It’s definitely not some sort of Deepak Chopra-woo. Serious scientists have seriously debated this issue—one of the more famous ones being Max Tegmark. Indeed, he’s proposed it as a way to test the many worlds interpretation, since it’s the only issue on which it gives different predictions from other mainstream interpretations.
This wouldn’t help: there’s a nonzero chance that somewhere in the universe, at some point, some particles spontaneously assemble into a copy of yourself. Now, it depends somewhat on your philosophical predilections of whether you consider a perfect copy of yourself with contiguous experience to yours ‘you’, but that’s nothing but the old Star Trek teleporter argument—if you’re happy being teleported, you should be happy being ‘resurrected’.
Likewise, the heat death won’t stop you: there’s always a nonzero chance that a local fluctuation in entropy provides enough usable energy to carry on for just an instant longer. As long as that’s true, there’ll always be a branch in the wave function where you continue to exist.
But the conclusion of the quantum immortality argument still can’t be correct. For one, there’s a simple way to kill any given observer: say |Ψ0> is the state of the universe before they were born, and |Ψ1> is the state of the universe today. Then, both are related by a unitary transformation: |Ψ1> = U|Ψ0>. But then, since unitary evolution is deterministic, there exists a unitary transformation U-1 such that U-1U = UU-1 = Id (the identity transformation), and hence, we only need to apply U-1 to the entire universe at the current moment, and get U-1|Ψ1> = |Ψ0>; in other words, a future state of the universe where the observer is absent. Hence, the logic of the quantum immortality idea can’t be correct: there are ways to remove a conscious observer from the universe. That much must be clear, simply because there are ways to create conscious observers (some of them quite fun), and physics, at least if the MWI is correct, is overall time-reversal invariant—whatever you can do, you can also undo.
Now, it’s quite hard to undo these things, to the great chagrin of many who thought that forgetting a condom just this once won’t be such a big deal. The above example involves recoherence: the rejoining of branches that previously split apart. This is a fantastically unlikely process—but it does have a non-zero likelihood of occurring. And the quantum immortality idea depends on the notion that the probability of surviving can be arbitrary small; but any non-zero likelihood of death dooms the quantum immortality argument: it’s not the case that there will always be a surviving copy of the observer.
However, this still means that you might expect a quite long life, narrowly escaping death again and again. But there’s no reason to believe that the ‘re-setting’ of the universe is the only way to snuff out somebody for good. If consciousness can ‘fade in’, with a new child being born, there must exist some mechanism for it to ‘fade out’—in the extreme, just taking that particular individual brain through whatever steps it took to become conscious in the reverse. Consciousness isn’t on-off, isn’t a binary thing; we can be less conscious now than we were just the moment before (if you doubt that, various substances are available for experimentation; I recommend a good scotch).
It’s a sort of sorites paradox: when does a conscious being cease to be conscious? And I don’t think there’s any hard-and-fast line. And that means, there’s also no hard-and-fast point at which consciousness must travel down another branch, so to speak. But there will, eventually, be a point where there’s no conscious experience to speak off anymore. So there will, eventually, be a point where any branching off is too late. Hence, while there’s no cul-de-sac, no circumstances such that death is guaranteed by some contrivance, there are roads to nowhere—continuous descents into states of ever-diminishing consciousness, where at some, unsharply defined point, there just isn’t any you within you anymore.