From this post it is written that "orientation preserving conformal maps respect CR. The matrix of the map must be a constant multiplied by some matrix of rotation that has a positive determinant. When this matrix is seen as the Jacobian of some function, you shall be able to show that the function respects CR."
I am not sure how to show this. Any help would be great. I think it has to do something with approaching a point parallel to one of the axis...