The answer would be a huge "NO". Why? the key word is "clone". A common misconception is that a "clone" would be 100% identical to the original. However, that is not the case at all. During the cloning process, many steps involve "mixing" of DNA strands that cannot be controlled, which leads to mutations. So mutations WILL occur even though you are cloning identical genes. So the clone will never be 100% identical to the original. Just find some photos of monozygotic twins (twins from a single zygote, pretty much what would happen during lab cloning), you will find differences when you look long and hard enough. Even more interestingly, these twins typically have distinct personalities, suggesting that their genetic make-up is different.
Additionally, do we consider 2 identical natural-born twins the same people, or 2 distinct individuals? They are absolutely 2 separate individuals. So even IF we can get a clone that is 100% identical to the original, we still need to consider them 2 separate individuals.
All in all, as long as we generate 100% biological beings, either the natural way or in the lab, we have to consider each of them distinct individuals. No question about it. It does not matter if they share all your memory. This is would be like you have a twin sibling who has been with you ever since both of you were born. Just because a person shares memory with you, it does not make him you.
The only option to "extend" your life is to make a robot and download your memory to it. Even that, this new bot can never be the 100% same as you. Since it is absolutely impossible to design a robot that not only looks like you, but also shares identical personality, etc. As long as it is designed by human, there will be differences. so it is IMPOSSIBLE to make a 100% clone of a human. When that is the case, the clone (be it 100% biological or mechanical) should always be considered a separate individual.
It's interesting that you entertain the possibility with a robot, but reject it with a clone. Even if a clone has minor genetic differences, would it not be a lot easier and faithful to re-imprint a neural pattern on a clone than on a robot?
I guess the clone setup might have distracted from the real issue, which is, can a stream of consciousness carry on in a different physical body, provided that the underlying physical structure of the consciousness (in this case, the brain and its neural pattern at a specific moment) is faithfully copied?
If we turned the concept around, and assume that there exists a self-aware AI with a similar stream of consciousness as real human beings, I think it's pretty easy to accept that you can just copy the AI onto another computer and have it exist again. However, would it still be the same AI?
Another note on twins vs clones. In my particular scenario, I feel that clones differ from twins in a fundamental way.
Although twins are born genetically identical, they are not necessarily born self-aware. Psychologically, we still don't understand how infants learn self-awareness, but it does seem to be acquired after birth. So twins would be two separate streams of consciousness because their consciousness developed separately. On the other hand, in the clone scenario, I was talking about somehow imprinting this consciousness instead of allowing the clone to develop its own.
Of course, this could very well be impossible. How do you modify a brain's neural patterns? It might be feasible if it's just a pattern of neural signals, then you could stimulate individual neurons into firing at a specific rate until you get the desired result. However, what if it requires a rearrangement of cells? Then that becomes an impossible task.
But then if we come back to the whole "downloading your memory into a robot", if we assume that it can be done almost flawlessly, does your stream of consciousness then get carried over? Sure a computer is different from an organic brain, but people who suffer brain damage are essentially working with a "different" hardware as well, and it would be hard to argue in those cases that their previous stream of consciousness was terminated.
What do you guys think?