What have you done?

You can find the explicit formula for your mapping using the cross ratio. You can also more easily find its inverse and then find the transformation itself.

After you've done that, both the real axis and the imaginary axis are circles in the wide sense, so they get mapped to circles in the wide sense as well. You know where are mapped to so that gives you the real axis. The imaginary axis is orthogonal to the real axis, so you can figure it out from the explicit formula as well.