f[z_] := z^2 + c;

c = .47; d = (1 + Sqrt[1 + 4 Abs[c]]) / 2.0;

n=9; 

Ax = Plot[0, {x, -d, d}, PlotRange->{-d, d}, AspectRatio->1]

Circ= Show[Graphics[Circle[{0, 0}, d]], AspectRatio->1]

Do[
 Fig[k]=ContourPlot[Abs[Nest[f,x + I y, k]] - d, {x, -d, d}, {y, -d, d},
	Contours->{0}, ContourShading->False,
		PlotPoints->70],
{k,0,n}]

Do[Show[Fig[k]], {k,0,n}]

all= Join[{Ax, Circ},Table[Fig[k], {k,0,n}]]
Show[ all]

Julia[c_,n_]:=Module[{d,Ax,Circ,Fign,f},d=((1+(Sqrt[1+4 Abs[c]]))/(2.`));f[z_]:=(z^2)+c;Ax=Plot[0,{x,-d,d},PlotRange->{-d,d},AspectRatio->1,DisplayFunction->Identity];Circ=Show[Graphics[Circle[{0,0},d]],DisplayFunction->Identity];
Fign=ContourPlot[Abs[Nest[f,x + I y, n]] - d, {x, -d, d}, {y, -d, d},
	Contours->{0}, ContourShading->False,
		PlotPoints->70];
Show[Ax,Circ,Fign,DisplayFunction->$DisplayFunction]];

Julia[1.05 I,7]