Is there another 3rd axis of imaginary numbers?

Hey r/math, please entertain a non-mathematician here. I've been trying to look this up, but have had no luck in finding anything--even speculation--on the subject.

Quite simply, since imaginary numbers can be thought of as extending into a second dimension perpendicular to the set of integers, should there not be a third dimension of "imaginary-imaginary numbers"? (as well as a fourth, fifth, etc...)

If these imaginary-imaginary numbers do exist, can I see a hyper-Mandelbrot set please? :D

(sorry for being such an ignoramus)

Edit: Wow, don't really have time to reply until after work today, but consider my mind blown!