No, without being able to define what consciousness is we can't make an experiment to test for it. The simplest explanation in my view is that qualia are a base "element" of the universe and information systems and our consciousness is a loosely drawn subset of qualia associated with our brain/body's information system. To be clear this is not dualism, the qualia are not some magical thing riding on top of physics and cannot change the path of particles from their physically determined routes.
Regarding quantum immortality I don't know if it necessarily demands a universe in which some form of every person lives forever by some unexpected chance of quantum improbability. At least in terms of a continuity of normal life it may be that the probability of a person living longer than a human body physically can is zero. However some have argued there is a non-zero chance that random quantum fluctuations could lead to brief instances of a person arising from "nothingness" anywhere and anytime in the universe, and some of those persons could by pure luck happen to match the mind of a person elsewhere. However this does start getting into the realm of silliness and quantum randomness and infinite universes allowing for literally every possible combination of particles.