Has someone ever tried to extend the complex number to $i^x=-1$?

I recently watched a series of videos about the history of complex numbers, and as I got to the part about their geometrical representation, I started to get more curious about the square in $$i^2=-1$$.

Indeed, we all know that the complex plane is made of two orthogonal spaces (Real and imaginary numbers), and I find it almost too neat that multiplying by i is essentially applying a rotation by $$90°$$ (this gives that the "angle" between the two spaces has to be $$90°$$)

My question is, has someone ever tried to see what would happen, and what kind of mathematical theories we could end up with, by analysing another set of numbers defined by $$i^x=-1$$? (x would be a real number, positive or negative). I'm guessing then the two spaces would not be orthogonal anymore, but there could also be a lot of other unforeseen consequences.

• Well, provided a definition of $z\mapsto z^x$ on suitable portions of $\Bbb C$, essentially one already ends up having a lot of solutions to the equation $z^x=-1$...
– user239203
Feb 18 '21 at 13:32
• I would suggest looking up roots of unity and cyclotomic fields - this is close I think to your intuition. And indeed, the study of these does yield interesting results!
– JMP
Feb 18 '21 at 13:41
• Ooh I learned about roots of unity a few years back but I had totally forgot about it. Never heard about cyclotomic fields though, I'll go check it out thanks ! Also z↦z^x makes me think of the mandelbrot set, I think I could also look in this area Feb 18 '21 at 13:44
• You "get" the complex numbers from $i^2=-1$ by imposing the usual ring axioms. How do you intend to "get" anything from $i^x=-1$? Do you want a topological $\Bbb R$-algebra with the ability to power anything to any real number? Not even complex numbers have that (in a well-defined way). Do you just want to be able to power elements by $x$? If so, what's stopping someone from arbitrarily declaring any multpilicative function "powering-by-$x$", other than restricting to such on the reals? Feb 19 '21 at 8:28
• The only way you'll get a meaningful answer is if you're very explicit about what assumptions you're imposing / what properties you desire. My guess is the answer will always be boring or degenerate (trivial). Also, how are quaternions at all relevant? Feb 19 '21 at 8:31

Also you can see some videos on YouTube for the same. Usually like complex numbers are intuitively used to solve signal or circuit problems where current and voltage are some phase apart, quaternions are used to solve many problems in the Robotics domain specially the POSE problem (position + orientation). Roll Pitch and Yaw are the $$3$$ rotations about X,Y and Z axes which can be combinedly representated as a single quaternion. Hope this helps...