Because the domain and range of a complex function are each two dimensional, the graph - the set of points (z, f(z)) - is four dimensional. Using the notation f(z) = f(x+iy) = u+iv, the graph is the set of 4-tuples (x,y,u,v). f(z) can rotate the graph in R4 (and project it onto the two-dimensional screen).
Start with the image of a veritcal strip of height 4π and map it onto an annulus under the exponential. Since the height is 4π, the annulus is "covered" twice.
Depending on how you rotate the graph, you should be able to see an exponential horn, sine curves, and even the Riemann surface of the logarithm. Think about the relationships between the graphs of f and f-inverse, and the Riemann surfaces for both.
