Can we make a coordinate system with two imaginary axes?

Question

As question suggests, can we make a coordinate system with two imaginary axes instead of one real and one imaginary (as usual for a complex plane)? Is it possible? And if yes does it have any meaning or it is just useless? Thank you!

Answer 1

It depends on what you mean with „coordinate axis“ and „imaginary“.

Sure, you can define a vector space of dimension three, let’s say $\Bbb R^3$ consisting of triples $(x_1,x_2,x_3)$ with $x_1,x_2,x_3 \in \Bbb R$ and componentwise addition. No one hinders you in writing those triples in the form $x_1 + x_2i + x_3j$ where we introduced two imaginary numbers $i$ and $j$ to separate the components.

It becomes interesting if your questions is about having a useful product on this three-dimensional vector space, which is compatible with the addition. If you want this multiplication to behave reasonably well (say being associative, distributive and having multiplicative inverses) then it becomes a theorem (many mathematicians tried to find exactly this for a long time…) that you cannot. So you might want to have a look at Frobenius Theorem