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.

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    $\begingroup$ It just all goes down to definitions. "Conjugate" has no fixed meaning for each and every extension of real numbers, and definitions of "conjugate" may be sometimes inconsistent and depend only on context, e.g. in $\Bbb Q(\sqrt{2})$ the conjugate of $\sqrt{2}$ is usually taken to be $-\sqrt{2}$, while when we talk in the context of complex numbers all real numbers are their own conjugates. $\endgroup$
    – Wojowu
    Commented Mar 24, 2016 at 10:11