Note that the circles t → exp(i*t) in the domain are mapped to exp(i*t) + c*exp(4i*t)
in the range. We recognize them as cycloidal curves. (that is, curve traced by rolling a circle on another circle.)
In this case we have three zeros of the derivative. But the behavior of the map near
these critical points is the same as in the earlier cases (near the zeros of their derivative).
Observe an important fact of complex analysis: No interior point of the domain is mapped to
the boundary of the image. In other words: The boundary of the image set consists only of
images of boundary points. No fold lines as in the non-conformal example can occur.