It's certainly a pleasure to see some working and a clear statement of where you're stuck.

Note that since the roots are not real, k > 1.

Real part: , k > 1 .... (1)

Imaginary part: , k > 1 .... (2)

From (1): .... (3)

Substitute (3) into (2) to get the Cartesian equation of the locus (it's a sideways parabola).