Complex numbers are extensively used in electronics so I never had a feeling of it as something purely synthetic, but most of n-dimensional math was for me beyond my capabilities to understand it any level higher then treating at as a purely mathematical abstraction of our 3d world - and thus all I'd do was just crunching the equitations.

Probably if I was exposed to real world applications of that math it would be easier to reason about it. I found that to be the biggest problem with my academic curriculum - there was a lot of abstract math concepts that they teach you in first 2 years, and you discover their actual usefulness only years later.