A lot of SF stories have been written along those lines. Lots of debate about how you would go about copying the mind, how much of the mind exists in the physical brain and how much is "virtual machine" or metaphysical and intangible. Of course there's the ultimate question, of whether a hypothetical brain-upload running on software would be "the real you" or merely an imitation; that is, whether there would be a continuity of conscience, or a transfer of the soul. I'm strongly inclined to think "not".
There was a time when the heart was thought to be the recepticle of the soul, and that the soul departed when the heart stopped. Open-heart surgery and heart transplants pretty much KO'd that theory. The idea of the brain being the link between the physical body and the intangible soul makes more sense to me, but then it begs the question of brain damage, brain surgery, trephining where part of the gray matter is skimmed away to relieve concussion pressure. People have survived and gone on to live with some degree of normalcy with a substantial chunk of their brain gone. Perhaps there is some specific portion of the brain, without which the consciousness cannot survive, that might be the that metaphysical link. Of course, its all a moot point to those who don't believe in an immortal soul, but I'd still think that the question of continuity of consciousness would be an issue to those folks.
If such a "mind upload" should become available (far fetched imho), I don't think I'd be intrested.
G.