Can the distance ($r$) or angle ($θ$) of the Polar coordinates contain Complex numbers ($a+bi$)?

Question

1. Is it possible that distance ($r$) or angle ($θ$) contains Imaginary or Complex number?
2. If the answer is yes, how can I convert a number like that (Polar with complex argument) to Rectangular number?
   For example: $(r,θ) = (5+2i, 3+4i)$ how to convert to $x+yi$ ?
   
   Thank you.

Answer 1

Hint

Substitute $i=e^\frac{i\pi}{2}$ and use the exponentiation rules for $r$, and the hyperbolic functions for the angle, along with some of the trignometric identities. This will give you two polar coordinates which then need to be added together and simplified.

Answer 2

First of all, if the distance of some complex number (x+iy) from the origin is complex, then it follows that either one of x or y is non-real, otherwise r cannot be complex. Thus your complex number cannot be constructed.

Secondly, complex angles do exist, but the notion of a complex argument of a complex number doesn't seem quite right, not in 2D Argand plane at least.