I think that part of the difficulty in comprehending consciousness from a scientific perspective is precisely because consciousness is an internal and personal experience. One way of coping with this difficulty has been by sidestepping the problem of dealing with internal experience altogether, as is done in the approach of behaviorism.
While being concerned with observable behavior is certainly a valid approach, denying the reality of internal experience, simply because we don't currently know how to observe it objectively, is unwarranted. For me, this internal experience under discussion is in fact my total experience. To me, it makes no sense to say that my personal experiences are unreal, simply because others can't experience them. I am therefore left with the conclusion that the primary aspect of my own experience is currently beyond the ability of science to comprehend, and similar statements apply to everyone else.
That said, I know of nothing about this primary aspect of my experience that I would expect to remain forever beyond the grasp of science. My total experience supports the assumption that scientific theories can one day be developed to account for consciousness. However, my experience also supports the assumption that such theories would be revolutionary, in no small part because of their scope. I therefore have an expectation that a successful theory of consciousness will have novel content that does not resemble that in any scientific theory in use today.
This brings me back on topic to make the following point. The conscious experience of dying is necessarily a personal experience, and discussing that personal experience scientifically suffers from at least the same difficulties that discussing all other personal experience does. Provisional estimates of the way consciousness relates to the physical world support the reasonable hypothesis that consciousness must stop at death because nervous activity ceases, but how can we test this hypothesis scientifically?
The film Brainstorm (1983)
focused on a scientist's quest to replay his colleague's personal experience of dying. Taping personal experience and sharing it, as if replaying a movie on a VCR, as done in the film, would provide an ultimate standard by which we might agree that we have scientifically comprehended at least an important part of the physical nature of personal experience. Though there are some indications that machines such as those depicted in the film might someday be possible, I don't think we're yet ready for anything like that.
Short of this Holy Grail of taping and replaying personal experience, how could we scientifically test the hypothesis that consciousness ceases at physical death?