Can we make a coordinate system with two imaginary axes?

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!

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