More fundamental than thinking is experience itself.
Regardless of whether there is a "you" or if it's some amalgamation of state that is loosely bounded together and "fooled" into thinking it is a unity, something is
there experiencing. At least in my frame there is.
This isn't something you can prove because it comes any sort of structure capable of doing proving. It's just something that's a given and you start from there.
Descartes' "Meditations on First Philosophy" is the originator of this idea. While it is dated, the form of its principal argument hasn't changed.
With regards to conscious unity, there is at least a weak form of it in the sense that you can't experience others' experiences. While it is possible that your own experience may not be fully unified, it is (very likely) disjoint from others' experiences.
You can experience another person's experience when you see them smile or cry. We call it empathy in modern parlance. The hogan twins joined at the head have an even more direct connection to each others' experiences:
I've never been very moved by ideas that I can't know or share other people's feelings, or other fanciful ideas like their blue is my green. It's more reasonable to assume they are like me because we share similar hardware (DNA) and software (Culture). Others hands look like mine, more or less. Others legs are like mine, more or less. And so others perception of green is like mine, more or less.
Telling me that my experience is prime, or fundamental doesn't tell me much. Similarly, saying I think therefore I am doesn't tell me much. What then am I and what is existence? I think therefore I am only as much as I think I am. And sometimes I forget myself.
Regardless of whether there is a "you" or if it's some amalgamation of state that is loosely bounded together and "fooled" into thinking it is a unity, something is there experiencing. At least in my frame there is.
This isn't something you can prove because it comes any sort of structure capable of doing proving. It's just something that's a given and you start from there.
Descartes' "Meditations on First Philosophy" is the originator of this idea. While it is dated, the form of its principal argument hasn't changed.
With regards to conscious unity, there is at least a weak form of it in the sense that you can't experience others' experiences. While it is possible that your own experience may not be fully unified, it is (very likely) disjoint from others' experiences.