Let us imagine lightning striking you so hard it disintegrates all your atoms and then, simultaneously, lightning strikes just a few meters from where you stood, somehow happening to rearrange all the atoms as you were before you disintegrated. The question now is did you really 'shut off' and then 'reappear' in the second instance? Or did you shut off and never reappear? Or did you experience two different realities? What if the lightning struck 10 different places, each time rearranging your body? Will you have 10 different perspectives of one body or will there be ten different bodies each with its own perspective ?
You might like reading about Parfit, his Reasons and Persons.
For what it's worth, I think a "person" stands in a direct causal relation with any further instance of them, and is not just an identical stucture of entities. This is also true to early Buddhism: as no two identical "persons" can ever exist from moment to moment.
I think I recall that Parfit says that "continuity" is given by the strongest causal relation between two moments. Why that ends up as as personal continuity, could perhaps be explained in terms of 'Buddha nature' -- our own place in the world. The mysterious question being why consciousness arises and ceases: not how it can be "unitary".