Suppose you have equation involving a number $s$

$s^2+ 1 = 0$,

to solve it one needs to treat $s$ as complex number $s = \pm i$, and introduce $i$ as imaginary unit.

Now suppose you have equation involving a number $s$ and its complex conjugate $\bar{s}$

$s \bar{s} + 1 = 0$,

What is reasonable approach to extend number system to solve this equation?

For example, is it ridiculous to treat $s$ as 'supercomplex' and introduce 'supercomplex' unit.