f[x_]=(x^3)-1;
g[x_]=x-((f[x])/(f'[x]));
redroot=N[1];
greenroot=N[-(1/2)+(((Sqrt[3]) I)/2)];
blueroot=N[-(1/2)-(((Sqrt[3]) I)/2)];
redlist={};
greenlist={};
bluelist={};
nr=(1000^2);
sqrtnr=(Sqrt[nr]);
Do[
	p[0]=-1.5+((3 k)/sqrtnr)+(-1.5+((3 l)/sqrtnr)) I;
	If[N[p[0]≠0],
			  finish=False;
		     n=0;
		     While[!finish,
			          n=n+1;
			          p[n]=N[g[p[n-1]]];
			          redtest=Abs[p[n]-redroot]<0.5;
			          greentest=Abs[p[n]-greenroot]<0.5;
			          bluetest=Abs[p[n]-blueroot]<0.5;
			         finish=redtest||greentest||bluetest;
			  ];
		    If[redtest,AppendTo[redlist,p[0]]];
		    If[greentest,AppendTo[greenlist,p[0]]];
		    If[bluetest,AppendTo[bluelist,p[0]]]],
	{k,0,sqrtnr},{l,0,sqrtnr}]

redpoints=
	Table[{Re[redlist[[k]]],Im[redlist[[k]]]},{k,1,Length[redlist]}];
greenpoints=Table[{Re[greenlist[[k]]],Im[greenlist[[k]]]},{k,1,Length[greenlist]}];
bluepoints=
	Table[{Re[bluelist[[k]]],Im[bluelist[[k]]]},{k,1,Length[bluelist]}];

redplot=ListPlot[redpoints,PlotStyle->{Red,PointSize[0.001]},PlotRange->{{-1.5,1.5},{-1.5,1.5}}];
greenplot=ListPlot[greenpoints,PlotStyle->{Green,PointSize[0.001]},PlotRange->{{-1.5,1.5},{-1.5,1.5}}];
blueplot=ListPlot[bluepoints,PlotStyle->{Blue,PointSize[0.001]},PlotRange->{{-1.5,1.5},{-1.5,1.5}}];
rootplot=ListPlot[{{1,0},{-(1/2),((Sqrt[3])/2)},{-(1/2),-((Sqrt[3])/2)}},PlotStyle->{Black}];
Show[redplot,greenplot,blueplot,rootplot,AspectRatio->Automatic]