F[x_, c_] = (x^2)+c;

Plot[{F[x,2],F[x,0],F[x,-2],x},{x,-5,5}, PlotRange->{{-5,5},{-5,5}}]

Plot[{F[x,0.1],F[x,0],F[x,0.5],x},{x,-5,5}, PlotRange->{{-5,5},{-5,5}}]

Plot[{F[x,1/4],x},{x,-5,5}, PlotRange->{{-5,5},{0,5}}]

Table[Nest[F[#,1/4] &,0.55,k],{k,0,35}]

Table[Nest[F[#,1/4] &,0.501,k],{k,0,70}]

ContourPlot[{(x^2)+c==x},{c,-5,5},{x,-5,5},ContourStyle-> {{Red, Thickness[0.01]}},PlotRange-> All, Axes->True, Frame->False, AspectRatio->1, AxesLabel->{"c","x"}]

Solve[(x^2)+c==x,x]

2 x /. x->(1/2) (1-(Sqrt[1-4 c]))

N[%/.c->-1]

2 x /. x->(1/2) (1+(Sqrt[1-4 c]))

(1/2) (1-(Sqrt[1-4 c]))/.c->-0.7

Table[Nest[F[#,-0.7] &,-0.48,k],{k,0,70}]

(1/2) (1-(Sqrt[1-4 c]))/.c->-0.76

Table[Nest[F[#,-0.76] &,-0.5,k],{k,0,3000}]

ContourPlot[{((((x^2)+c))^2)+c==x},{c,-5,5},{x,-5,5},ContourStyle-> {{Red, Thickness[0.01]}},PlotRange-> All, Axes->True, Frame->False, AspectRatio->1]