You see, the argument you've made makes more sense in a world that, in some eastern disciplines, is referred to as "nondual". The problem with dualistic thinking -- thinking there's an ego in the skin, a barrier between me and you, a cosmic king, etc -- all depend upon the idea of division to the point that everything stands alone, that everything is fixed, that everything is static. Your example highlights the illusoriness of that quite well: where is the fixed self in an unfixed form? There's no "there" there. You might be arguing that it means there IS a "there" there, but that's because you may be arguing for an unchanging special snowflake process.
Now, what may remain is consciousness without a self, sometimes called "not-self", an awareness of the contents and what it can objectify, but itself is not the contents. Even one of the responses to this remark with the "nervous system" is still defining a self to a part of the body; it's still linking a "me" to a thing. There's no you in any conceivable way that we have identified it to be in thoughts, which is to say there's no "thing" that is truly me, for the me is always in ideas, in images, in concepts.
How can you save an image? What concept truly lives outside of mind? If we can say that there's an awareness that exists prior to the ego, and prior to thoughts that arises in awareness, you're more "that" than anything else, but even describing it this way is to say you're once again a thing or a particular feature or particular process, and none of that is true.
