We first need to clear the widespread confusion between consciousness and mind.
It is the fact that the word "consciousness" is terminally ambiguous which led some philosophers to come up with the notion of subjective experience and qualia. We will follow this welcome innovation here.
We can define the mind as a sort of subjective representation of the world. Our mind is essentially qualia we take to be things in the so-called real world. Thus, we should distinguish the qualia from what qualia represent. We can see qualia as information, so that the mind has an information content. This information can concern anything, from memories of past events, to current percepts, thoughts, sensations, intuitions etc.
The most plausible and therefore reasonable assumption is that this information is somehow codified and stored by the brain, presumably through neuronal connections.
The crucial consequence of this is that the whole information content of our mind is stored in the brain, which also means that this information is quickly and irremediably destroyed when the person dies.
What might not disappear upon death is subjectivity itself, although this is totally speculative. Further, even if this is true, without the brain, there is nothing left to experience for subjectivity, at least nothing remotely similar to experiencing what it is to be a living human being.
It might also be that subjectivity is somewhat like space, ready to experience the information content of any living brain. This, however, is also totally speculative. Further, the information content associated with brains would still disappear with them, so that we would be unable to remember past subjective experiences.
These last two suggestions are possibilities in the sense that current science is not in a position to disprove them, but that does not prove that they are true. We just don't know. Further, even if they are true, they wouldn't amount to anything comparable to any of the various religious ideas about life after death.
The idea is that we can easily distinguish two things: the quality of our subjective experience and the information content of what we experience. The information content is really what specifies who we are at every instant of and throughout our lives. This is what most plausibly totally disappear upon death. The rest, subjective experience, or "consciousness" properly understood, would remain, but as a sort of blank screen. Not much to look forward to.