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Plot Complex Roots

    
Visualize complex roots using the absolute value of complex numbers:
Wolfram Language code: pts = ComplexExpand[z /. Solve[z ^ 3 - 27 I == 0]]
Wolfram Language code: Show[ComplexContourPlot[Abs[z ^ 3 - 27 I], {z, 4}, Contours -> 15], ComplexListPlot[pts -> pts], ImageSize -> Medium]
Visualize complex roots using the phase of complex numbers:
Wolfram Language code: Show[ComplexContourPlot[Arg[z ^ 3 - 27 I], {z, 4}, Contours -> 15, ColorFunction -> "Rainbow"], ComplexListPlot[pts -> pts], ImageSize -> Medium]