Julia Set

c = -0.2546715638093696` - 0.6936732634062556` I;
n = 12;
step[ps_, t_] := With[{ps2 = Sqrt[ps - c]},
   Which[
    t < 1, (* shifting *)
    ps - t c,
    1 <= t < 2, (* square rooting *)

    Map[#^(1 - 0.5 (t - 1)) &, ps - c],
    2 <= t <= 3, (* duplicating *)
    Join[ps2, ps2 + -2 (t - 2) ps2]
    ]
   ];
topts[ps_] := Map[{Im[#], Re[#]} &, ps];
p0 = Join[
   Table[0.7 (Cos[\[Theta]] + I Sin[\[Theta]]), {\[Theta], 0, 2 Pi,
     0.1}], Table[
    0.2 (Cos[\[Theta]] + I Sin[\[Theta]] + 0.8 + I), {\[Theta], 0,
     2 Pi, 0.2}],
   Table[0.1 (Cos[\[Theta]] + I Sin[\[Theta]] - 3 + I), {\[Theta], 0,
     2 Pi, 0.4}]];
ps = NestList[step[#, 3] &, p0, n];
Manipulate[
 Graphics[
  {Darker@Darker@Blue,
   Point[topts[step[ps[[Floor[t]]], 3 Mod[t, 1, 0]]]]},
  PlotRange -> 2, Background -> RGBColor[0.8, 0.9, 1.0]
  ],
 {t, 1, n + 0.99}]