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The real contour plot is taken from a textbook (Quantum Optics by Girish Agarwal, Cambridge U Press, 2013, page 80):

The real contour plot is taken from a textbook (Quantum Optics by Girish Agarwal, Cambridge U Press, 2013, page 80):

Adding Contours -> Range[-0.35, 0.35, 0.1] + 0.03 to my code generated the following plot, which is better, but another problem happened: the colors in the legend are not as they should be. For example, we have the same color for (-0.12,-0.02) and for (0.28,0.38)

The real contour plot is taken from a textbook:

The real contour plot is taken from a textbook (Quantum Optics by Girish Agarwal, Cambridge U Press, 2013, page 80):

To emphasize that it is not a problem for a special case, here I have added another code and figure. You can see the "ugly" ovals are definitely incorrect.

ContourPlot[
1/((2 + 2/E^16) \[Pi])
2 (E^(((2 + 2 I) - x - I y) ((-2 + 2 I) + x - I y)) + 
E^(((-2 - 2 I) - x - I y) ((2 - 2 I) + x - I y)) + 
E^((-x - I y) (x - I y)) Cos[4 Im[(2 - 2 I) (x + I y)]]), {x, -4, 
4}, {y, -4, 4}, PlotLegends -> Automatic, 
ColorFunction -> "Rainbow", FrameLabel -> {"x", "y"}, Frame ->  True,
FrameStyle -> Directive[Black, 18], PlotPoints -> 180, 
PlotRange -> All, PlotRangePadding -> 0]