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I would like to have the ticks on the imaginary axis of a ComplexListPlot to be labeled with $-2i$,$-i$,$0$,$i$,$2i$ instead of $-2$,$-1$,$0$,$1$,$2$.

I have already tried it with the Ticks option like it is suggested in this question, but then I got the error

A tick position in the value of the Ticks option should be a number.

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We can use the undocumented function Charting`FindTicks[] to generate ticks whose tick labels we can alter to be multiples of $i$, like so:

g1 = ComplexListPlot[{-2 - 2 I, -1 - I, 0, 1 + I, 2 + 2 I}];

yticks = MapAt[If[NumberQ[#], I #, #] &,
               Charting`FindTicks[{0, 1}, {0, 1}] @@
               Last[Charting`get2DPlotRange[g1]], {All, 2}];

Then,

Show[g1, Ticks -> {Automatic, yticks}]

list plot with complex ticks

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