Bundle Theory suggests that consciousness is not a mystical property of the mind, one that exists outside of the body and is not subject to the same stresses and existences as the body. Budle theory, instead, suggests that the mind and consciousness are in fact a by-product of our brain being active. The exact nature of this 'by-product' is more open to debate, and there is much disagreement as to what causes this experience.
To a transhumanist, the question of conciseness becomes very important, what am I allowing to exist if not myself? Consciousness is fundamentally who you are, and the experiences which you live, if you did not have this property of self-awareness, you would not be aware of your actions. So naturally, to a transhumanist, keeping this state of self-awareness continuous is critical, whatever it's origins. This, of course brings up a very important question relating to the concept of 'uploading' or 'transferring' your consciousness.
If I were to take an exact picture of my brain and all of it's neruons, and recreate or simulate this configuration on a computer, we would have a simulated 'me' which would have similar memories and a similar mind, allowing variance for the exact properties the simulation accounts for regarding hormones, neurotransmitters and other biological affectors. However, there is one simple problem which many either do not see, or do not consider a problem: The simulated copy of myself is simply not me.
To the simulated copy, it would seem the process worked, and that I had uploaded myself onto a computer and continued my existence. However, to the original biological me, it would seem as nothing has changed, and this is the me that we are really concerned with. The biological me is the same continued self-awareness as who started the procedure, leading only a copy being created, and not an uploading or transferring. The fact that the biological me can still exist is one of the fundamental problems with this concept. This problem is encountered in what I call the 'teleportation problem'.
If I were to use a teleportation device which broke apart my body, and reformed an exact copy at the other end with a different set of atom (a common concept for teleportation), we are left with two fundamental problems. The first, is that no continued consciousness transferred in the procedure, and the question of: Why bother destroying the original?
If we are to assume that this method would be possible, why bother destroying the original set of atoms which steps into the device, why not instead, simply create a copy? There is no feasible way in which the property of self-awareness I transferred over, and instead a new self-awareness is formed. Some bundle theorists argue that it doesn't matter, because our self-awareness I nothing more that a combination of properties which align to form our mind, a combination which the 'clone' then replicates. But I would argue, that this is precisely why it's a problem.
If we assume that our consciousness has no special property, and is simply an side effect of our mind being active, then recreating the same conditions will not create the same self-awareness, only a copy of it, and a copy isn't good enough.
This is a common problem encountered in the philosophy of consciousness, one which has been plaguing all sorts of minds since time immemorial. I would argue, that the only thing that matters is continued consciousness, a property of broken self-awareness. The real question in my mind, is under what conditions is this continuous self-awareness broken. It is possible, that it is broken when our mind shuts down, or even under anaesthetic. However, it is also possible that we are perfectly conscious under anaesthetic, it's just a matter of us not being able to act, or to remember. This is, of course, backed up by claims and observations of dreaming while under anaesthetic.
But regardless of the exact conditions the continuation is broken, we are likely never to know, which is why we must be conservative and careful in our actions. It is very possible the the continuation doesn't even exist and that the 'me' writing this is simply not the same as the one that will post it. This is a very real possibility to deal with, but one we can never know.