Personally I think the cool part is that the complex logarithm takes the stereographic projection and turns it into the Mercator projection. That is, it conform

Personally I think the cool part is that the complex logarithm takes the stereographic projection and turns it into the Mercator projection. That is, it conformally maps the plane onto an infinite two-ended cylinder. The expression log(–1) = iπ more or less just tells you how to normalize the coordinate system on the cylinder. If you wanted you could pick a somewhat different coordinate system. This one is conventional and often convenient though.