Also available as an interactive applet (scrollwheel or pinch to zoom so you can adjust the axis how you want).
Given: a complex number C plotted on the complex plane.
Bisect the angle formed between C, the origin O, and the positive X (real) axis.
Extend OC past O by a unit line segment to U.
Find the midpoint, M, between U and C.
Construct a half-circle centered at M through U and C.
Draw the perpendicular to UC through O to intersect the half-circle at I.
Draw a circle centered at O through I.
The circle intersects the bisector at S.
S is a square root of C.