I would like to create a contour plot of $e^z-\dfrac{z-1}{z+1}$ that looks like the picture on the left, but can only manage the one on the right. I have tried incrreasing the number of contours, but this doesn't have any effect. Am I missing something fundamental?
Show[Table[ContourPlot[{Im[E^(x + I y) - ((x + I y) - 1)/(x + I y) + 1] == k,
Re[E^(x + I y) - ((x + I y) - 1)/(x + I y) + 1] == k}, {x, -4, 4}, {y, -20, 20},
ContourStyle -> {Directive[{Red}], Directive[{Blue}]},
PlotPoints -> 100, Contours -> 20, AspectRatio -> Automatic], {k, -10, 10, 2}]]
The plot I am trying to recreate is from this website (specific PDF).