I imagine that a clone is made of me. (Feel free to imagine yourself as "me" here.) Not an ordinary clone that simply has the same DNA as me, though, but an exact clone where every particle in his body and brain has the same relative position and momentum as a corresponding particle in me, and vice versa. If you believe in additional non-material aspects of the self such as a soul, then assume for the purpose of this thought experiment that an exact copy is somehow made of those as well. My clone is exactly the same as me in every conceivable way, and is therefore even initially thinking the same thoughts as I am.
This brand new person would have all of my memories stored in his brain, and therefore he would believe that he has lived my life. He would have the profound belief that he is the real me, and that I am an imposter that somehow suddenly appeared. He would feel that he has existed as a conscious being continuously from his (my) earliest memory (except for being interrupted daily by sleep). His feeling is of course an illusion, because he was created only very recently.
Now suppose that I vanish and my clone continues living. Everything would be the same as if I had continued living and had never had a clone, and my clone would have the same internal experience of being me as I myself would have had. Becuase my clone's belief of continuous existence is an illusion, even though the resulting state of the universe is just as if I myself had continued living, this suggests that my own experience of a continuous mental existence is similarly an illusion. At any moment, my sense of having lived my life arises from the memories that exist in my brain at that moment, whether "real" or fake, and not from the actual history of events that may have produced those memories.
It therefore appears to be more accurate to say that my mental self exists only at the present moment, perhaps for the time it takes to "think one thought", such as "Now what was it that I was just about to do?" My mental self then immediately ceases to exist and is replaced by a new one at the next moment who believes that it has existed continuously over the lifetime of the body that each of us inhabited very briefly.
One might argue that an absolutely identical clone's mind is in fact a continuation of the single mind that is me, and so its belief of a continued existence is not an illusion, and so neither is mine. But if my own body continued to exist along with the clone, then clearly my clone might be exactly like me but would not be me. I certainly wouldn't agree to be killed just because I can see a copy of my mind continuing onward in another body. I don't think I would agree even if I saw that a copy of my mind was continuing onward in a potentially immortal robot's body, even if I had to choose between the robot "living" and me living.
One might then argue that I and my clone have each had a continuous mental existance for our entire perceived lives, though we previously were the same mind and now are separate minds or beings that will diverge henceforth. And so, again, neither of us is imagining our continuous mental existence. But suppose that the clone were not a copy of me at all, but were created as a new adult with a completely contrived set of memories installed into his brain. His impression of having lived for a few decades would then irrefutably be an illusion. Yet he has precisely the same sort of contrived existence as a clone of me would have, except without there happening to be another person just like him. So a clone of me has the same illusion of a mental past as a totally fabricated human would.
One might then concede that my clone's impression of having lived a continuous mental life is an illusion, but still argue that my life is fundamentally different because my mind has remained in the same body, and so my belief is not an illusion. But regardless of our two histories, our present states are identical. We both believe that we have existed continuously only because of the memories that are stored in our brains. In addition, human bodies are continually fluctuating assemblages of particles that come and go. My body has gradually come to consist of a different set of particles than when I was younger, just as my clone's body was created with a different set of particles all at once. And quantum physics has shown that our most basic particles are really more like energy or probability distributions anyway.
So my illusion is as real as my clone's. My hypothetical decision mentioned above to continue living in my body rather than allowing a copy of my mind to live much longer in a robot is therefore irrational. Yet I don't think I would change my decision no matter how convinced I am intellectually of this argument, because the illusion is so powerful.
When our bodies eventually die, the only experiential difference will be that no further momentary mental selves will appear when the previous ones cease to exist as usual. There is no need to worry about death, because we are already dying at each moment and being replaced by newer brief flickers of consciousness that are glad that they are "still" alive. The final vanishing will be no worse than all those before it.
There are no continuous selves, each in an individual body. There are only fleeting moments of awareness that bubble to the surface in all of the bodies that are carrying the stream of life that has arisen on the planet. We are all one process.