Here are my musings on the subject. I hope you find it useful.
The exponential map on complex numbers is similar to the Cartesian-polar change of coordinates in 2D:
Horizontal lines such as
will be mapped to rays extending from the origin, with the complex component
determining the angle of inclination. Vertical lines such as
will be mapped to circles centered at the origin. The real component,
will determine the radius of the circle. The imaginary axis gets mapped to the unit circle. (In fact, the exponential map is a covering map of the circle from a universal cover.)