>>53
You're not understanding what dimension means at all, I think that's your problem.
Just as you say "real numbers correspond to areas" in your complex plane, real numbers correspond to area in the normal cartesian plane, just you have negative areas instead of positive, and that pretty trivially happens because of you multiplying each axis by sqrt(-1).
You're calling an area something of dimension two, but it's not got dimension two if you just specify the magnitude of the area, it's like a vector, it's magnitude still has dimension one.
I can't really think of a clearer way to tell you that you're wrong.