This problem presupposes metaphysical realism, so you have to be a metaphysical realist to take the problem seriously. Metaphysical realism is a particular kind of indirect realism whereby you posit that everything we observe is in some sense not real, sometimes likened to a kind of “illusion” created by the mammalian brain (I’ve also seen people describe it as an “internal simulation”), called “consciousness” or sometimes “subjective experience” with the adjective “subjective” used to make it clear it is being interpreted as something unique to conscious subjects and not ontologically real.
If everything we observe is in some sense not reality, then “true” reality must by definition be independent of what we observe. If this is the case, then it opens up a whole bunch of confusing philosophical problems, as it would logically mean the entire universe is invisible/unobservable/nonexperiential, except in the precise configuration of matter in the human brain which somehow “gives rise to” this property of visibility/observability/experience. It seems difficult to explain this without just presupposing this property arbitrarily attaches itself to brains in a particular configuration, i.e. to treat it as strongly emergent, which is effectively just dualism, indeed the founder of the “hard problem of consciousness” is a self-described dualist.
This philosophical problem does not exist in direct realist schools of philosophy, however, such as Jocelyn Benoist’s contextual realism, Carlo Rovelli’s weak realism, or in Alexander Bogdanov’s empiriomonism. It is solely a philosophical problem for metaphysical realists, because they begin by positing that there exists some fundamental gap between what we observe and “true” reality, then later have to figure out how to mend the gap. Direct realist philosophies never posit this gap in the first place and treat reality as precisely equivalent to what we observe it to be, so it simply does not posit the existence of “consciousness” and it would seem odd in a direct realist standpoint to even call experience “subjective.”
The “hard problem” and the “mind-body problem” are the main reasons I consider myself a direct realist. I find that it is a completely insoluble contradiction at the heart of metaphysical realism, I don’t think it even can be solved because you cannot posit a fundamental gap and then mend the gap later without contradicting yourself. There has to be no gap from the get-go. I see these “problems” as not things to be “solved,” but just a proof-by-contradiction that metaphysical realism is incorrect. All the arguments against direct realism, on the other hand, are very weak and people who espouse them don’t seem to give them much thought.
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.