Theoretically speaking, how could we manage to live forever while remaining recognizably ourselves to us? One way is to ensure that the brain remains fully functional indefinitely. Another is to move the brain into a new and/or indestructible "container,” whether carbon, silicon, metal or a combination thereof. Not surprisingly, these notions have received extensive play in science fiction, from the messianic angst of The Matrix to Richard Morgan’s Takeshi Kovacs trilogy.
To give you the punch line up front, the first alternative may eventually become feasible but the second one is intrinsically impossible. Recall that a particular mind is an emergent property (an artifact, if you prefer the term) of its specific brain –- nothing more, but also nothing less. Unless the transfer of a mind retains the brain, there will be no continuity of consciousness. Regardless of what the post-transfer identity may think, the original mind with its associated brain and body will still die –- and be aware of the death process. Furthermore, the newly minted person/ality will start diverging from the original the moment it gains consciousness. This is an excellent way to leave a detailed memorial or a clone-like descendant, but not to become immortal.