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.
Yes. But not in this sense. Quaternions are usually termed as the extension of the complex numbers which was first discovered by William Hamilton. You can have a look here: https://en.wikipedia.org/wiki/Quaternion
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...