Think back to yourself when you were a darling child of five. Is the person you're thinking of the same being as the you who is reading this post?
How can you prove that the you who woke up this morning is the same being who went to sleep the night before, and not just a fresh manifestation of consciousness experiencing a continuity of memory and biology?
If some evil madman strapped you to a chair, sawed off the top of your skull and methodically removed your brain bit-by-bit as he engaged you in conversation, at exactly what stage would you consider yourself gone?
If the same evil madman cut out the section of your brain that stores your memories and replaced it with the memories of another victim, would the being who woke up in your body still be you?
I think it's reasonable to extrapolate that the idea of human personhood is no more or less a fiction than the idea of avatar personhood. Both conceptions are merely convenient labels that take a present-moment experience and hypostatize it forward and backward through imagined time. Doesn't this make the immersion vs. augmentation debate a moot point:
O Shariputra, all dharmas are marked with emptiness. They do not appear nor disappear, are not tainted nor pure, do not increase nor decrease. Therefore in emptiness: no form, no feelings, no perceptions, no formations, no consciousness; no eyes, no ears, no nose, no tongue, no body, no mind; no color, no sound, no smell, no taste, no touch, no object of mind; no realm of eyes...until no realm of mind-consciousness; no ignorance and also no extinction of it...until no old-age and death and also no extinction of it; no suffering, no origination, no stopping, no path, no cognition, also no attainment with nothing to attain. from The Heart Sutra