Julia2 =
Compile[{x,y,{m, _Integer}},
Module[{z = x + I * y,n = 0, k = 1.},
If[Abs[z] > 4, n = m];
While[n < m && k == 1, 
If[Abs[z + 0.1225599999871782`-0.7448599999870137` I] < Exp[-5], k = 0,
z = z^2 - .12256 + .74486 I] ; If[Abs[z] > 4,n = m];
n++];
k]
];

Timing[DensityPlot[Julia2[x,y,100], {x, -1.5,1.5}, {y, -1.5, 1.5}, PlotPoints -> 50, Mesh -> False]]

Timing[DensityPlot[Julia2[x,y,100], {x, -1.5,1.5}, {y, -1.5, 1.5}, PlotPoints -> 300, Mesh -> False]]

Timing[DensityPlot[Julia2[x,y,100], {x, -.4, -.2}, {y, .4, .6}, PlotPoints -> 300, Mesh -> False]]