2) Suppose . Equating real and imaginary parts gives |a|=|b| and ab=-1/2. The (two!) solutions are then and .
3) This is the set of points z whose distance from 1 is the same as their distance from -1. The locus of points equidistant from two points is a straight line perpendicular to the line joining the two points and passing through its midpoint. In this case, it is the line on the imaginary axis.