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I believe I would still be me. It would have been my body that had morphed into an atom-by-atom identical copy of your body but my soul would remain intact as I do not believe it would have similarly been affected. My thinking is that there would be two people whose bodies are identical but whose real selves are different.
Another tough question. As a Christian I believe that if this happened the bodies may be identical but the souls would not. Since a soul is something that is not a physical organ or body part one has to have faith to believe a soul exists. Without the soul the morphed identical body would not be me but rather an identical copy of you. Time travel is depicted as a body being transmitted back to the past or forward to the future and the nano particles come back together as the person that traveled through time. I do not understand the process since I am not an expert in physics but I do believe it will happen someday.
Similarly, consider two completely identical individuals existing separately in distinct realms. The surroundings within each realm would vary, thereby influencing the progression of each individual over a period.
If over the next thirty years, my body (including my brain) very slowly and steadily morphed into an atom-by-atom identical copy of your body (including your brain), I would still be me. I would not be you. I would still maintain my individuality in that case. We all are endowed with our unique selves which are eternal and imperishable. The uniqueness of my existence would be all the more conspicuous in the circumstances described above. All of this would occur only to prove that the core of my existence is unique to every other human being.
If the brain is also morphed into identical copy of you or someone else then I presume that the memories are also getting morphed. If the memories are morphed into a copy of someone else then I would not be me.
If I know this thing is happening and in next 30 years from now I will be someone else identical copy then I will try to make the someone like me or be like me so that I can reverse the effect and before morphed into that person I will make that person be like me.