Adding a root of $z\bar z=-1$ to $\mathbb C$ 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

A way to construct something like this is to consider the polynomial ring $\mathbb{C}[X]$, where conjugation is extended by imposing $\overline{X}=X$. 
Then consider the quotient $R = \mathbb{C}[X]/(X^2+ 1)$. This is a ring with unity, yet not a domain of course. 
Put differently, consider $\mathbb{C}^2$ with coordinatewise addition and conjugation, and multiplication given by $(a_1,b_1)(a_2,b_2) = (a_1a_2 - b_1b_2, a_1b_2 + a_2b_1)$. Identify the complex numbers with the elements $(c,0)$.  
