I made myself a little interactive with geometer's sketchpad so I could look at all solutions of (a + bi) + (c + di) = (a + bi)*(c + di)
The product of the distances of the two numbers from 1+0i is always 1. I didn't prove this.
The implication is that solutions to the above problem must have distance 1 from 1+0i. The solutions found in my other post are clearly the only ones that can.