>TL;DR: complex numbers are the unique algebraic extension to the real numbers The simplest, (or maybe most compact) extension. You can also achieve closure b

>TL;DR: complex numbers are the unique algebraic extension to the real numbers The simplest, (or maybe most compact) extension. You can also achieve closure by going from R to R^2 and adding the Geometric Product. This is also closed, but it includes vectors in R^2 as well as the complex numbers.