The complex numbers are the algebraic closure of the reals: it is a field extension that gives roots to all polynomials.
It can also be constructed abstractly: in the Cayley-Dickson construction from $\Bbb{R}$, the first step gives $\Bbb{C}$ then the second step gives $\Bbb{H}$, the quaternions. In the first step we lose the orderability of $\Bbb{R}$ as a field, yet gain algebraic closure.
Likewise, the next step loses commutativity. Is there some other operation (usefully) defined on $\Bbb{C}$ and $\Bbb{H}$, but is only closed on the latter?