I believe that human consciousness is entirely manifest in the brain (and, of course, it's effected by other nerves elsewhere in the body). But moreover, I believe that brain is performing "normal computation," by which I mean Turing complete computation. Because of that, there is no real reason why the same computation responsible for human consciousness couldn't run equivalently on other Turing complete hardware.
I could imagine how a suficiently advanced machine could simulate one or more consciousnesses.
What I can't imagine is how you, the person sitting in front of the computer will somehow wake up in the computer. I can only see copies waking up. Even if you are dead. For the people who love you, having a copy would be great, but you will still be dead.
I can't believe I'm bringing this up, but the Arnold Schwarzenegger movie "The 6th Day" really matches my views on the subject, and also matches your views.
(Obvious spoilers for the movie)
In the movie Arnold is cloned, wakes up later, and goes about to find out what's going on. At one point he finds out HE is the clone! Back to our discussion. From the point of view of the Copy, the procedure was successful - the consciousnesses was transferred to the machine and the copy continues existence. From his POV he does wake up in the computer after the procedur. For you, though, it's a copy that wakes up. Perhaps your brain is destroyed during the upload process and you stop to exist. For the copy you (that is, him) continue to exist and avoid brain death.
If a scientist asks you, "Was the upload successful?", you (the original) will answer "No". The copy you will answer "Yes".
For the digital copy to not notice the transition, there would probably have to be more going on—like a Matrix-esque simulation of the physical world or a realistic android to house the copy.
