You may be thinking of the Ship of Theseus:
https://en.wikipedia.org/wiki/Ship_of_Theseus

I was talking to a relative over Christmas holiday about this. We were talking about consciousness and how it is an emergent property of brains, much like digestion is an emergent property of stomachs and intestines. I was saying that I'd have no problem becoming a robot bit by bit as my broken bio parts are replaced by whatever high tech the future brings. As long as it happens bit by bit, I'll never lose conscious identity and I'll consider myself one entity from birth to this future robot version of me.

So I'd say yes, same boat/body/being. But operating in a different sea of information if we have upgraded our behavior and thought processes for better ones.

Cool stuff to contemplate! :)