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Oh man, I knew I had asked this question in the right place.
Thank you!
With imaginary numbers, I visualize something like a needle popping up and moving through cartesian space in new directions or dimensions in all sorts of unexpected and intricate ways, and eventually they find utterly extraordinary and elegant things like the Mandelbrot set. So I wondered if there are other "hacks" or "cheats" that open up new types of progressions and behaviors for study.
Someone else in the thread also mentioned Dirac doing something along the lines of (a)(0) ≠ 0 to handle some of the infinities that pop up in physics.