1
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{ComplexContourPlot[ReIm[z], {z, -3 - 3 I, 3 + 3 I}, PlotLabel -> z,  Contours -> {Range[-2, 2]}, ContourLabels -> All],  ComplexContourPlot[ReIm[f[z]], {z, -3 - 3 I, 3 + 3 I},  PlotLabel -> f[z], Contours -> {Range[-2, 2]}, `ContourLabels -> All]} // Grid[{#}, Frame -> True] &`

This code plots only the grid, not exp function

enter image description here

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3
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Expanding on Daniel Huber's answer,

Clear["Global`*"]

f[z_] := Exp[z]

Column[
 Row[#, Frame -> True] & /@
  Outer[
   ComplexContourPlot[#2[#1],
     {z, -3 - 3 I, 3 + 3 I},
     PlotLabel -> Style[#2[#1], 14, Bold],
     Contours -> {Range[-2, 2]},
     ContourLabels -> All,
     ImageSize -> 200] &,
   {z, f[z]}, {ReIm, AbsArg}, 1],
 Frame -> True]

enter image description here

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4
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Your drawing is a plot of {Abs, Arg}, not ReIm. Look at the following:

ComplexContourPlot[ReIm[Exp[z]], {z, -3 - 3 I, 3 + 3 I}, 
 ContourLabels -> True, Contours -> {Range[-2, 2]}]

enter image description here

ComplexContourPlot[{Abs[z], Arg[z]}, {z, -3 - 3 I, 3 + 3 I}, 
 ContourLabels -> True, Contours -> {Range[-2, 2]}]

enter image description here

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