On a more general note, any metaphysical notion, be it an interpretation of quantum mechanics, a general concept about other universes, is in its nature unscientific, can't be proven by science and probably can't be proven at all. Quantum mechanics does one thing – describes how our universe works within its own space and time. It can't be used to extrapolate what happens outside the universe, before the universe or after you die from your own perspective, because those "places" haven't been observed, can't possibly be observed and are very likely to differ so significantly in nature that anything you know is inapplicable to them at all.
1. Assume multi-world interpretation of quantum mechanics leads to quantum immortality for all individuals. For our universe to exist, let alone support life, the quantum fluctuations, their randomness and indeterminism in quantum processes need to be within precise constraints as defined by the laws of physics we know (or think we know). Any change in quantum fluctuations could destroy the universe, and if randomness stopped being random, I am not sure what will happen. And that's exactly what will be going on in your immortalverse. Of course, you could just be struck with an enormous streak luck winning the intergalactic lottery every Planck time turning into an eternal anomaly that lives against all odds and against the laws of physics, but this is profoundly more unlikely than finding yourself in a universe where your luck matches the local laws of physics. Living for a second that way has a probability that makes the Ackermann function sweat.
2. Existence itself is undefined for these metaphysical realms. Existence is the things that we can observe directly or indirectly. Nobody will observe your afterlife, nor will you observe anyone else's, so it doesn't even exist, how can we be talking about the more well-defined physical concepts? Worse, existence here is tied to probability – if six people saw a goose, there is probably a goose, because the likelihood of light shaped like a goose just randomly popping into six people's eyes is just incredible. Well, if the whole of your personal existence is more unlikely than that, how can anything exist in your realm?
I do believe in immortality of sorts that arises from much simpler things that have nothing to do with quantum mechanics or complex physics though:
- I think that we overestimate our personal importance, and that a person carrying out your legacy or mentally arriving at the same place as you accidentally is more than enough, and your personal uniqueness doesn't add much to that if at all, and you can share and immortalise your memories if you think they are this important.
- If you were immortal in the traditional sense, your life would start repeating itself. When it does, you're as good as dead, you'd be living the same finite thing over and over again.
- Living your life over and over is the same as living it once. Time is the natural progression from cause to effect that we observe, implying there's more to it is giving it metaphysical properties we have not observed. If the effect takes you back to the cause, that does nothing for the posterity of anything in-between, outside of our perception of time, which in the grand scheme of things is irrelevant. Been there, done that.