I get something vaguely similar with

DensityPlot[Sin[3 Arg[(x + I y)^3 - 1]], {x, -2, 2}, {y, -2, 2}, 
 ColorFunction -> "Rainbow"]

I don't quite understand it, though.

Yes, you can overlay a DensityPlot and your original plot with Show[plot1,plot2]. Be careful of the order: plot1 is at the bottom, plot2 is on top.

Here it is:

Show[DensityPlot[Sin[3 Arg[(x + I y)^3 - 1]], {x, -2, 2}, {y, -2, 2}, 
  ColorFunction -> "Rainbow", Exclusions -> None, PlotPoints -> 61], 
 ListPlot[Table[{Re[E^(I*2 \[Pi]*n/12)], Im[E^(I*2 \[Pi]*n/12)]}, {n, 
    0, 20}], AspectRatio -> Automatic, 
  PlotStyle -> {{ Black, PointSize[.02]}}],
 Table[ 
  Graphics [{Thick, Dashed, Black, 
    Line[{{Re[E^(I*2 \[Pi]*n/12)], Im[E^(I*2 \[Pi]*n/12)]},
      {Re[E^(I*2 \[Pi]*(n + 1)/12)], 
       Im[E^(I*2 \[Pi]*(n + 1)/12)]}}]}], {n, 0, 20}]]