c=-1;

Q[z_] = (z^2)+c;

Table[Nest[Q[#] &,0.,k],{k,0,35}]

c=0.13-0.45 I;

Q[z_] = (z^2)+c;

Table[Nest[Q[#] &,0.,k],{k,0,35}]

c=-0.03+0.85 I;

Q[z_] = (z^2)+c;

Table[Nest[Q[#] &,0.,k],{k,0,35}]

c=-0.43-0.05 I;

Q[z_] = (z^2)+c;

Table[Nest[Q[#] &,0.,k],{k,0,35}]

c=-0.9253+0.15325 I;

Q[z_] = (z^2)+c;

Table[Nest[Q[#] &,0.,k],{k,0,35}]

c=-0.7253+0.1325 I;

Q[z_] = (z^2)+c;

Table[Nest[Q[#] &,0.,k],{k,0,35}]

c=-0.03+0.85 I;

Q[z_] = (z^2)+c;

Pts = {{2.,1}};

wval = 2.+1.I

Solve[wval==Q[z],z]

wval = (z/.Solve[wval==Q[z],z][[RandomInteger[{1,2}]]])

For[k=0,k<10000,k++;
wval = (z/.Solve[wval==Q[z],z][[RandomInteger[{1,2}]]]);
If[k>150, Pts = Join[Pts,{{Re[wval],Im[wval]}},1];
]
]

NewPts = Drop[Pts,1];

ListPlot[NewPts,PlotStyle->{{Red}}, PlotRange->{{-2,2},{-2,2}}, Axes->None]

c=-0.9253+0.15325 I;

Q[z_] = (z^2)+c;

Pts = {{2.,1}};

wval = 2.+1.I

Solve[wval==Q[z],z]

wval = (z/.Solve[wval==Q[z],z][[RandomInteger[{1,2}]]])

For[k=0,k<20000,k++;
wval = (z/.Solve[wval==Q[z],z][[RandomInteger[{1,2}]]]);
If[k>150, Pts = Join[Pts,{{Re[wval],Im[wval]}},1];
]
]

NewPts = Drop[Pts,1];

ListPlot[NewPts,PlotStyle->{{Red}}, PlotRange->{{-2,2},{-2,2}}]

ListPlot[NewPts,PlotStyle->{{Red}}, PlotRange->{{-.2,.2},{-0.85,-0.4}}, Axes->None]

c=0;

Q[z_] = (z^2)+c;

Pts = {{0.,0.}};

C2 = Max[Abs[c],2];
For[x=-2,x≤ 2,x=x+0.01,
For[y=-2,y≤ 2,y=y+0.01,
v=x+I y;
StopVal=0;
For[k=1,k<75,k++;
If[Abs[v]>C2, {StopVal=1,Break;},v=N[Q[v]]];
]
If[StopVal==0,Pts = Join[Pts,{{x,y}},1];]
]
]

ListPlot[Pts,PlotStyle->{{Red}}, PlotRange->{{-2,2},{-2,2}}, AspectRatio->1]

c=-1-0.67 I;

Abs[c]

Q[z_] = (z^2)+c;

Pts = {{0.,0.}};

C2 = Max[Abs[c],2];
For[x=-2,x≤ 2,x=x+0.01,
For[y=-2,y≤ 2,y=y+0.01,
v=x+I y;
StopVal=0;
For[k=1,k<75,k++;
If[Abs[v]>C2, {StopVal=1,Break;},v=N[Q[v]]];
]
If[StopVal==0,Pts = Join[Pts,{{x,y}},1];]
]
]

ListPlot[Pts,PlotStyle->{{Red}}, PlotRange->{{-2,2},{-2,2}}, AspectRatio->1, Axes->None]

Table[Nest[Q[#] &,0.,k],{k,0,35}]

Pts = {{2.,1}};

wval = 2.+1.I

Solve[wval==Q[z],z]

wval = (z/.Solve[wval==Q[z],z][[RandomInteger[{1,2}]]])

For[k=0,k<20000,k++;
wval = (z/.Solve[wval==Q[z],z][[RandomInteger[{1,2}]]]);
If[k>150, Pts = Join[Pts,{{Re[wval],Im[wval]}},1];
]
]

NewPts = Drop[Pts,1];

ListPlot[NewPts,PlotStyle->{{Red}}, PlotRange->{{-2,2},{-2,2}}]