My two cents: in the discussion with the red squares, you should specify the sign of your imaginary number. Here, for having your examples to work, one needs a positive sign.
Yeah I know... this is picky ;)
Informal answer. The symmetry of the complex plane makes it difficult to perform operations which would have been simple in the ordinary plane. The Euler identity in a sense exemplifies this since it does something very simple geometrically while the equations are quite complicated.
This actually shocked me. In physics, more symmetry makes things easier. :)
Good job in spotting that mistake! Thanks. I made adjustments. To avoid talking about signs I identified it to a part of the imaginary axis.
Well there is of course lots of nice math that you can apply on the complex plane which makes use of the underlying symmetry (like everything related to singularities). Once you know this math things get easier :)
I hope you will continue posting. My preferred application is the residue theorem. This is a so wonderful piece of math! :)