Is There a Higher Structure $\mathbb{C}$ is a Part of?

Question

The reals $\mathbb{R}$ and the set $i\mathbb{R}:=\{ir\vert i^2=-1, r\in \mathbb{R}\}$ are subsets of $\mathbb{C}$. By extension, the structures $(\mathbb{R},*)$ and $(i\mathbb{R},*)$ (for some binary map $*:A\times A \to A$) are sub-structures of $(\mathbb{C},*)$ So, is there some higher set or mathematical structure $\mathbb{C}$ is a subset of aside $\mathbb{C}^n$?