I'm posting this response to try to get input from other tutors....
It seems to me that there will be no solution to the given equation. I will be interested in anything useful that others have regarding this problem.
Below are my thoughts about the problem that lead me to my conclusion that there is no solution.
Represented in polar form, x+iy and y+ix have equal magnitudes and are symmetrical about the line y=x.
The angles associated with the two complex numbers can then be represented as (45+a) and (45-a) degrees.
But applying deMoivre's Theorem to the given expression shows that no value of a will produce a value for the given expression with an angle of 90 degrees -- as required by the given expression being equal to 2i.
Experimenting with random values of a on wolframalpha.com in fact shows that any value of a produces a result in the form where c is a real number.
