The complex logarithm is a multivalued function because it has an imaginary part that depends linearly on the argument of the independent variable. In terms of a radial parametrization of the independent variable, one has
As this argument increases or decreases infinitely, the elevation of the imaginary part winds up or down around the vertical axis in an infinite corkscrew. On the principal branch the argument is the inverse tangent of the ratio of the imaginary and real parts of the independent variable. Due to the cyclic nature of the exponential, it changes by 2π when moving between Riemann sheets, so that for integer n one has in general
Visualizations of the imaginary part of the complex logarithm use a variety of colors for the surface. One choice that is appealing for its simplicity and colorful result is to set the hue from the argument of the independent variable, i.e., coloring by elevation:
While this is a pleasant looking visualization, it does not convey enough information about the complex structure of the function. Coloring by the argument of the complex logarithm is more informative but so strikingly different that it requires a bit of thought to figure out why it differs so much from elevation coloring.
For a given point on the complex plane, the argument of the complex logarithm itself is
Its asymptotic behavior is independent of whether the value of the radius starts above or below unity, as can be seen with this interactive plot:
For radial values above unity the initial argument is zero, corresponding to the color red because it has a hue value of zero in the HSL color space. To keep hue values in the range of zero to unity the argument needs to be divided by 2π when assigning colors. For radial values below unity the initial argument is π, which corresponds to a hue of one half or cyan.
What is curious is that for large positive values of the argument of the independent variable, the asymptotic value of the argument of the complex logarithm itself is one half of π, regardless of whether the argument starts from zero or π. This corresponds to a hue of one quarter, which is a shade of green. Conversely, for large negative values of the argument of the independent variable, the asymptotic value of the argument of the complex logarithm itself is minus one half of π, again regardless of whether the argument starts from zero or π. This corresponds to a hue of minus one quarter, which is a shade of purple.
What this means is that for coloring by argument there will be some interesting structure on the principal branch of the imaginary part of the complex logarithm,
but the coloring quickly becomes monochromatic on sheets farther from the origin:
All logical and yet somewhat bizarre...
The real part of the complex logarithm does not have the multivalued structure of the imaginary part, but one can still color by the corresponding sheet of the imaginary part:
This just emphasizes that the bulk of the complex structure occurs on the principal Riemann sheet.
Since the absolute value incorporates the multivalued nature of the imaginary part it too is multivalued. It looks like this:
Again, the complex structure becomes noticeably simpler for large values of the argument of the independent variable, apart from the symmetrically interleaved Riemann sheets.