Depends upon what you mean by “consciousness.” A lot of the literature seems to use “consciousness” just to refer to physical reality as it exists from a particular perspective, for some reason. For example, one popular definition is “what it is like to be in a particular perspective.” The term “to be” refers to, well, being, which refers to, well, reality. So we are just talking about reality as it actually exists from a particular perspective, as opposed to mere description of reality from that perspective. (The description of a thing is always categorically different from the ontology of the thing.)
I find it bizarre to call this “consciousness,” but words are words. You can define them however you wish. If we define “consciousness” in this sense, as many philosophers do, then it does not make logical sense to speak of your “consciousness” doing anything at all after you die, as your “consciousness” would just be defined as reality as it actually exists from your perspective. Perspectives always implicitly entail a physical object that is at the basis of that perspective, akin to the zero-point of a coordinate system, which in this case that object is you.
If you cease to exist, then your perspective ceases to even be defined. The concept of “your perspective” would no longer even be meaningful. It would be kind of like if a navigator kept telling you to go “more north” until eventually you reach the north pole, and then they tell you to go “more north” yet again. You’d be confused, because “more north” does not even make sense anymore at the north pole. The term ceases to be meaningfully applicable. If consciousness is defined as being from a particular perspective (as many philosophers in the literature define it), then by logical necessity the term ceases to be meaningful after the object that is the basis of that perspective ceases to exist. It neither exists nor ceases to exist, but no longer is even well-defined.
But, like I said, I’m not a fan of defining “consciousness” in this way, albeit it is popular to do so in the literature. My criticism of the “what it is like to be” definition is mainly that most people tend to associate “consciousness” with mammalian brains, yet the definition is so broad that there is no logical reason as to why it should not be applicable to even a single fundamental particle.
MWI only somewhat makes sense (it still doesn’t make much sense) if you assume the “branches” cannot communicate with each other after decoherence occurs. “Quantum immortality” mysticism assumes somehow your cognitive functions can hop between decoherent branches where you are still alive if they cease in a particular branch. It is self-contradictory. There is nothing in the mathematical model that would predict this and there is no mechanism to explain how it could occur.
Imagine creating a clone which is clearly not the same entity as you because it is standing in a different location and, due to occupying different frames of reference, your paths would diverge after the initial cloning, with the clone forming different memories and such. “Quantum immortality” would be as absurd as saying that if you then suddenly died, your cognitive processes would hop to your clone, you would “take over their body” so to speak.
Why would that occur? What possible mechanism would cause it? Doesn’t make any sense to me. It seems more reasonable to presume that if you die, you just die. Your clone lives on, but you don’t. In the grand multiverse maybe there is a clone of you that is still alive, but that universe is not the one you occupy, in this one your story ends.
It also has a problem similar to reincarnation mysticism. If MWI is correct (it’s not), then there would be an infinite number of other decoherent branches containing other “yous.” Which “you” would your consciousness hop into when you die, assuming this even does occur (it doesn’t)? It makes zero sense.
You see the issue right here, you say the reality in which you survive, except there would be an infinite number of them. There would be no the reality, there would be a reality, just one of an infinitude of them. Yet, how is the particular one you find yourself in decided?
MWI is even worse than the clone analogy I gave, because it would be like saying there are an infinite number of clones of you, and when you die your cognitive processes hop from your own brain to one of theirs. Not only is there no mechanism to cause this, but even if we presume it is true, which one of your infinite number of clones would your cognitive processes take control of?