cjc = Compile[{{power, _Integer}, {c, _Complex}},Module[{imax=1000, dzmax = 2500., maxpt = 10^5, z, dz, roots, branches, i = 2, res = 0. + 0. I, pt = 0},
branches = Table[Exp[-2. N[Pi] I k/power], {k,0,power - 1}];
z = Table[0. + 0.I,{imax}];
dz=Table[1. ,{imax}];
roots=Table[1,{imax}];
Map[{Re[#], Im[#]}&,NestWhileList[
Function[
z[[i]]=branches[[roots[[i]]]](z[[i-1]]-c)^(1/power);dz[[i]]=power Abs[z[[i]]]^(power-1)dz[[i-1]];
res = z[[i]];
If[i<imax&&dz[[i]]<dzmax,
i++;
roots[[i]]=1
(* else *),
While[i>1&&roots[[i]]==power,
roots[[i]]=1;i--];
roots[[i]]++];res],
0.+0.I ,Function[(i > 1) && (pt++ < maxpt)]]]], RuntimeOptions->"Speed"];
Graphics[{PointSize[0.0001],Point[cjc[3,-0.040000000000000036` + I * -0.78`]]}, AspectRatio->1]

Manipulate[Graphics[{PointSize[0.0001],Point[cjc[p,d[[1]] + I d[[2]]]]}, AspectRatio->1], {{p, 3}, Range[10]},{{d,{-.5,-.05}},{-2,-2},{2,2}},SaveDefinitions->True]