I get how to construct number systems: $\Bbb{N,~Z,~Q,~R,~C}$. But what about the hypercomplex? How do we go from $\Bbb C$ to $\Bbb H$ and $\Bbb O$?
I've also heard it stated that the octonions are the "center" of mathematics. If that's true, then what would be a good way to explain or depict that graphically?
I took the liberty of adding the crucial part of the interview within the linked video in the question body to make it more easy to understand the part where the OP said that he heard "the octonions are the 'center' of mathematics".
Eric Weinstein:
"If I were running the NSF I would say why are we not putting money into there four very strange objects called $F4$, $E6$, $E7$ and $E8$ to try to figure out what they are telling us. We know, that they are at the center of mathematics" [emphasis by me]