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

CPts = {{0,0}};
For[cxval=-2, cxval≤1, cxval=cxval+0.01,
For[cyval=-1.5, cyval≤1.5, cyval=cyval+0.01,
cval = cxval + I cyval;
If[Abs[Q[NestWhile[Q[#,cval]&,  0., Abs[Q[#,cval]]< 2&,1,200],cval]]<2., 
CPts = Join[CPts,{{cxval,cyval}},1];
]
]
]

MandPlot1 = ListPlot[CPts,PlotStyle->{{Red}}, PlotRange->{{-2,2},{-2,2}}, AspectRatio->1]

CardPlot1 = ParametricPlot[{(1/2)(Cos[θ]-(1/2)Cos[2θ]),(1/2)(Sin[θ]-(1/2)Sin[2θ])},{θ,0,2π},PlotStyle->{Blue,Thickness[0.005]}];

CircPlot1 = ParametricPlot[{(1/4)Cos[θ]-1,(1/4)Sin[θ]},{θ,0,2π},PlotStyle->{Green,Thickness[0.005]}];

Table[Nest[Q[#,-1.1+0.12 I] &,0.,k],{k,0,35}]

Show[{MandPlot1,CardPlot1, CircPlot1}]

BPts = {{0,0}};
For[k=-2.0, k≤ 0.25,k=k+0.0025,
{y = 0.0;
For[r=1,r<600,r++,
{y=Q[y,k];
If[r>550, 
BPts = Join[BPts,{{k,y}},1];
];}
]}
]

BifPlot1 = ListPlot[BPts,PlotStyle->{{Black}}, PlotRange->{{-2,2},{-2,2}}];

Show[{MandPlot1,CardPlot1, CircPlot1,BifPlot1}, ImageSize->{800,800}]