Mathematicians disagree on the essential structure of the complex numbers (2024) 5 months ago
The link is about set theory, but others may find this interesting which discusses division algebras https://nigelvr.github.io/post-4.html
Basically C comes up in the chain R \subset C \subset H (quaternions) \subset O (octonions) by the so-called Cayley-Dickson construction. There is a lot of structure.