This half-serious question is inspired by the answer to my previous one, Want something like Cayley formula for unitary matrices
The equation $z^2=-1$ does not have solutions in $\mathbb R$; adding a solution produces $\mathbb C$.
The equation $z\bar z=-1$ does not have solutions in $\mathbb C$; adding a solution produces what?
Update - having learned more thanks to the comments and the answer, I've now posted a question on MO with hopefully more serious and interesting content in it: https://mathoverflow.net/q/248241/41291