It's fun to twist thought experiments around and see what kinds of strange conclusions you can get out of them. Let's do that with quantum immortality. I'm hopefully going to convince you to be absolutely terrified of the many-worlds interpretation of quantum mechanics.
Surely you've heard of the quantum suicide thought experiment. It's the one where you prove to yourself that the multiverse exists by stepping in front of a loaded gun and having it fire through your gray matter if a quantum coin flip comes up heads. The idea is, if you repeat that 100 times, you won't die because as soon as you die, your experience stops, so the only possible outcome, for you, is that the gun didn't fire. If you're worried about being alive between when the bullet leaves the barrel and when it strikes your head, replace the gun with a nuclear bomb or insta-kill radiation going at the speed of light. Sure, in about 2100 parallel universes, everyone else sees you die, but in at least one, the coin will come up tails 100 times in a row, and that's what you'll experience. It's an event so unlikely that you'll be forced to conclude the multiverse exists. Don't try this at home.
If you think about the quantum suicide experiment a little bit more, you'll realize that the quantum coin coming up tails isn't the only way to survive. A flaming meteor could tunnel through your ceiling and destroy the death machine moments before it kills you. Or, a little more likely, the gun would jam, or an earthquake would tip it over, etc. Quantum weirdness can make these "classical" life-saving events happen, too. In other words, the quantum coin wasn't really an important part of the thought experiment. Even if the gun was set to always fire, something would save you. If you're about to get hit by a truck, there's some universe where a freak sinkhole opens up and swallows the truck before it hits you. That's quantum immortality.
So, quantum immortality is the crazy idea that you can't ever die because some crazy quantum fluke will always save you. Sounds like a pretty good deal, but is it really something you'd want? Let's think about it.
Remember that objection to quantum suicide, "What about your experience after the gun fires but before you die?" That turns out to be really important. After the bullet clears the other side of your skull, you'll probably still be experiencing something. I mean, it's going to take a while, at least a few milliseconds, for all of your brain activity to stop. This is how we turn quantum immortality into quantum hell: Why should quantum mechanics save you before the gun fires, and not after blood starts draining out of a 9mm hole in your skull?
To bring more intuition into this version of the thought experiment, let's imagine you're dying a slow death. Every day, your condition gets worse, and after years of deterioration, you finally get pronounced dead. In at least one world, quantum immortality isn't going to let you really die. As you get closer and closer to true death, rarer and rarer quantum flukes happening inside your body keep you going. You simply approach death, never attaining it. It's like there's a quantum healing force keeping you just barely alive.
If this isn't the most horrifying thought experiment ever conceived, please tell me what is. If quantum mechanics is accurate, the many-worlds interpretation is right, you being alive implies there's at least one future with non-zero amplitude where you remain alive, and the quantum suicide argument is valid, then this will happen to you!
It might not be as bad as it sounds. Things you'd rather not experience, like bad thoughts and pain, probably need a lot of brain infrastructure to work. Quantum healing probably won't sustain those structures for long. And so, eventually, you'll be experiencing almost nothing, yet still something. You will degrade to a point where maybe once every millennium you will feel a mild blip of experience, and then you will be out again for another thousand years. Weird.
Thought experiments are fun; don't take them too seriously.