My chief problem is the perspective from which you look. I'll try to illustrate
Let's say that Minsky is bored, sitting in the Chinese Room and calculating the life of a some AI, with inputs/outputs coming from a simulated world he's also calculating. Then Minsky realizes he feels weird, and then he remembers that he's dead, so he stops running the AI on pen and paper.
Now the AI is frozen in time. From the view of the AI's consciousness, from the inside, this moment is lasting for eternity - even though he is not dying, the state he was left in allows for many next moments to come; though they never do as nobody touches Minsky's papers. The papers remain and contain the consciousness of the AI, allowing it to perceive the moment for eternity from the inside, while people out there can perceive the papers and also the conscious AI within them.(Searle has an objection to what I am saying, but he is chased off buy angry tumblrinas with sexual harassment allegations.)
The same story could play out with Minsky finishing the life of the AI, and stopping the execution of its program right in its last moment, where the AI's perception of time slows down and since there is no consciousness to perceive the next moment, it will keep perceiving the last one for eternity. Now not only is the execution of his program stopped, but its current state means that there's no next step to calculate. Still, the papers are there containing this last state of the program, and so both from inside the program's consciousness, and from outside (Minsky looking at the papers), the moments remains for eternity. This however requires the papers to be there and hold it.(Searle is mad now, but the tumblrinas are many and he has to leave campus altogether to design some conscious robots to exterminate them out of their own free wills, which won't get him charged for murder.)
The two important things about the pen-and-paper AI above is that 1) the moment for eternity is possible because of the papers that contain it, and 2) because the papers were dedicated to the mind/brain of this AI. In the case of a human, the brain is right here and will continue to change during and after the last conscious moment, thus changing that last conscious moment. Even though it was the last one perceived from within the consciousness, it is tampered with, changed from the 'outside' during the falling apart of the fleshy brain creating it. Furthermore, unlike in the AI case, the whole program (reality) is not dedicated to the mind of that single dying human; thus the _point of view_ that is his consciousness is not separate from the world it emerged from, and that world keeps moving on. There won't be a paper, memory dump or snapshot holding the life of this person, there won't exist a point of view that perceives his last moment either from the inside, or from the outside. Both of these get changed by the world in which he took place.
It is you (who thinks about the dying person) taking him out of the context of reality, and turning him and his consciousness into a discrete set of moments (and corresponding states of his body), then discarding the unconscious parts after death, and giving extra significance to his inner perception of time that turns this last moment into unchanging and eternal. It is a theory, a model in the shape of a closed system you place on the person, who is not actually separated from the rest of the world in such a way, and is actually an open system.