I would not want immortality as such, and that would be impossible anyway. There is a finite, albeit unimaginably long, lifespan on even the elementary particles of the Universe. You'd run out of any energy sources to keep yourself animated long before that, and stop having anything we'd recognize as "nature" to appreciate long before that. The sun will consume the Earth in five billion years, but the Earth will be stripped of its atmosphere and become uninhabitable much sooner than that, on the order of a billion years.
The storage requirements to even remember what you were for such a length of time become staggering. You end up with a post-biological species that becomes a plague on the Universe, greedily converting every bit of matter it can find into near-ideal-computing-device just to maintain its ever-expanding sense of self.
I would, however, like more of a say in the particulars of my own end, and given the choice of going digital or reaching a humiliating and incapacitated wasting conclusion that burdens my family, I would rather have the former. I wouldn't want immortality, but a bit more existence might be nice.
The "it's a copy" problem is a little thorny, but only given certain approaches. Imagine, for instance, a type of nanomachine that can live in the brain and identify neurons faltering due to age, and replace that neuron with a fully functional non-biological substitute. Your consciousness is never interrupted or "transferred" to another device, but just replaced one negligible bit at a time with something that doesn't decay and die. Would you ever stop being you? At what point?
Another idea that is possibly more plausible is that this could happen through a brain-computer interface that allows humans to augment their natural brains with a technological expansion. As the hardware portion becomes cheaper and more powerful, it becomes more plausible that this device could be designed to sustain the consciousness autonomously when the natural brain to which it is attached stops functioning.
However, I seriously doubt this will ever really become possible, and if it did I have some truly grim ideas about what would happen socially as a consequence.