I used to think this until I thought about, instead of an instant transfer, gradual integration with electronics. First an implant that connects your brain to a computer interface. Then one that gives you extra neural capacity.

Your thoughts and memories move back and forth between biological and digital substrate. Over time you get more and more digital capacity and less biological until one day you are all computer. There would be no point where you go from one to the other. It is the ship of Theseus.