Let $p = -1+2i$ and $q=1-2i$

The perpendicular bisector is the set of points equidistant to points $p$ and $q$. The distance between some point, $z$ and $p$ is $|z-p|$. Similarly, The distance between some point, $z$ and $q$ is $|z-q|$.

Thus the set of points equidistant to both $p$ and $q$ is given by the equation:

$|z-p| = |z-q|$

Another way to do this is solving for the equation of the line in x-y coordinates first. The points are $(-1, 2) and (1, -2)$, the midpoint is $(0, 0)$ and the slope is $m_1=-2$. Thus the perpendicular bisector has the slope $m_2 = -\frac{1}{m_1} = -\frac{1}{2}$. Therefore, the perpendicular bisector has the equation:

$Re[(-\frac{1}{2} + i)z]$