I'm not a mathematician and don't play one on TV, so I'm usually happy to settle for an approximate answer. It looks as if (-1.185379618i + 2.5308348106) squared is very close to -6i + 5. And I didn't even have a group to avoid!

What bugs me about trying to work with complex numbers and other weird **** is that I find I can easily misunderstand or misapply the "rules." For me at least, there's usually no way to check my answer against something else, like a physical object, that's exists independently of those rules. If I square my answer and get -6i + 5 that's a good sign, but I'm still not 100% sure that I've correctly followed all the applicable rules for squaring complex numbers.

It's different when I'm playing with "my kind" of problem -- say, something like a set of data points meant to represent a 3D map of a landscape. If I also have a photo of the same landscape, I can calculate where in the photo each landmark should appear. When (as often happens) some of my data points don't fit the picture perfectly, I get to reexamine my data points, tinker with my estimate of where the picture could have been taken from, and try again.

I'm not sure how manic I've ever gotten (probably not very), but knowing that I can successfully pull off a bunch of intricate calculations and have the result agree with other, independent information says to me that I could very well still be making sense.