p[z_,alpha_,maxn_]:=Module[{r,t,w},r=Abs[z];t=Arg[z];
w=r^alpha*Exp[I alpha t];
Table[{Re[w Exp[I n alpha 2 Pi]],Im[w Exp[I n alpha 2 Pi]]},{n,maxn}]]

(* 1の1/8乗 *)

g8=ListPlot[p[1,1/8,8],AspectRatio->Automatic,PlotStyle->{PointSize[0.03]}]

(*1のルート2乗*)

groot2a=ListPlot[p[1,Sqrt[2],100],AspectRatio->Automatic]

groot2a=ListPlot[p[1,Sqrt[2],1000],AspectRatio->Automatic]

Manipulate[ListPlot[p[1,Sqrt[2],n],AspectRatio->Automatic],{n,1,1000}]