This also brings into question "Ship of Theseus". If we were to genetically modify all our cells, would we still be us? Who/What are we? Am I the same person that went to sleep or am I a new person with the same memories?

If we were to become immortal, assuming people wanted to, everyone could eventually become a master in every subject, so would it not be safe to assume we would find ways to change our genomes in response to changes?