Felem1[Z_]=((Z+(Sqrt[(Z^2)-4]))/2);  
Felem2[Z_]=((Z-(Sqrt[(Z^2)-4]))/2);
F[Z_]=Piecewise[{{Felem1[Z],Re[Z]>0},{Felem2[Z],Re[Z]<0}}];
ParametricPlot[{{Re[F[X+I Y]],Im[F[X+I Y]]}/.{Y->-0.3},
{Re[F[X+I Y]],Im[F[X+I Y]]}/.{Y->-0.2},{Re[F[X+I Y]],Im[F[X+I Y]]}/.{Y->-0.1},
{{Re[Felem1[(X/1.5)]],Im[Felem1[(X/1.5)]]},{Re[Felem2[(X/1.5)]],Im[Felem2[(X/1.5)]]}}/.Y-> 0,{Re[F[X+I Y]],Im[F[X+I Y]]}/.{Y->0.1},{Re[F[X+I Y]],Im[F[X+I Y]]}/.{Y->0.2},{Re[F[X+I Y]],Im[F[X+I Y]]}/.{Y->0.3}},{X,-3,3}]

g[z_]=(z+c + ((((1+c))^2)/(z+c)))Exp[I α]/.α-> -π/10;
ParametricPlot[{Re[g[Exp[I θ]]],Im[g[Exp[I θ]]]}/.c-> -.15,{θ,-π,π}]

c=-0.17-.1I
ParametricPlot[{{Re[g[F[(X+I Y)]]],Im[g[F[(X+I Y)]]]}/.{Y->-0.3},
{Re[g[F[X+I Y]]],Im[g[F[X+I Y]]]}/.{Y->-0.2},{Re[g[F[X+I Y]]],Im[g[F[X+I Y]]]}/.{Y->-0.1},
{{Re[g[Felem1[(X/1.15)]]],Im[g[Felem1[(X/1.15)]]]},{Re[g[Felem2[(X/1.15)]]],Im[g[Felem2[(X/1.15)]]]}}/.{c-> c},{Re[g[F[X+I Y]]],Im[g[F[X+I Y]]]}/.{Y->0.1},{Re[g[F[X+I Y]]],Im[g[F[X+I Y]]]}/.{Y->0.2},{Re[g[F[X+I Y]]],Im[g[F[X+I Y]]]}/.{Y->0.3}},{X,-2.3,2.3}]

ParametricPlot[{{Re[g[F[X+I Y]]],Im[g[F[X+I Y]]]}/.{Y->-0.3,c-> -.12+.1I},
{Re[g[F[X+I Y]]],Im[g[F[X+I Y]]]}/.{Y->-0.2,c-> -.12+.1I},{Re[g[F[X+I Y]]],Im[g[F[X+I Y]]]}/.{Y->-0.1,c-> -.12+.1I},
{{Re[g[Felem1[(X/1.15)]]],Im[g[Felem1[(X/1.15)]]]},{Re[g[Felem2[(X/1.15)]]],Im[g[Felem2[(X/1.15)]]]}}/.{c-> -.12+.1I},{Re[g[F[X+I Y]]],Im[g[F[X+I Y]]]}/.{Y->0.1,c-> -.12+.1I},{Re[g[F[X+I Y]]],Im[g[F[X+I Y]]]}/.{Y->0.2,c-> -.12+.1I},{Re[g[F[X+I Y]]],Im[g[F[X+I Y]]]}/.{Y->0.3,c-> -.12+.1I}},{X,-2.3,2.3}]