Eh, this still doesn't go beyond the fundamental "I think therefore I am." I can prove to myself that I have consciousness, but for everyone else, ¯\_(ツ)_/¯
You can't prove even that.
If you examine it closely, the argumnent only proves that "it" thinks.
"It" is not necessarily an "I", for what restricts the thinking to an I (i.e a part of reality)? It could be the whole reality that does the thinking as far as the argument goes.
It proves itself if you define "I" as the same as the experiencer of the thinking, but "I" (as many other complex word) is much more overloaded.
Unfortunately all our word definitions seems shaky, if we want to describe something that is the base requirement of those very definitions.
Maybe the best we can do is to deconstruct the above using more simple or base terms, but the meaning of those terms maybe also depends on the content of experience not the mere fact of experience:
I experience thought -> experience of thoughts exists -> experience exists -> something exists
So upon experiencing thought you may conclude that "something exists", or "there IS something"...
Of course you can. You know it to be true that you posses consciousness because you experience it directly. What is impossible (empirically) is knowing that about anyone or anything else.
We know there is thinking. There is no reason to believe that subject-object duality has any basis in reality, or that any individual, including our "self", has a sufficient delineation to consider it an independent entity.
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.