My tests in approaching four dimentional complex equations by solving for an imaginary/real ratio, and doing some inverse trig with the real and imaginary parts "where necessary" (It's more of an exception when it isn't as far as I've seen). I don't know if it all works, but all are (hopefully) algebraically solved by abusing complex geometry! I've found I can successfully integrate complex equations by solving for angles with respect to each part, deriving, rearranging, sometimes magic, and integrating to find "stuff". There's also plenty more, and I go on rants in the description of almost each one, because I'm completely insane and this is a cry for help. Turns out sin and cos are what hold on complex planes, even when they look identical to sinh and cosh. I abused this to the highest degree I could and produced (occasionally potentially provably true) complex planes. The best explanation for most of what I'm doing involves an imaginary-imaginary plane based on the derivitives of x and y, and... y, it's not that hard, just a bit out of the box. Anyone actually interested in this will find it extremely easy once you see the patterns, and (in my experience) impossible to truly explain.