F[x_,y_]:={x,y}+(({x,-y})/((x^2)+(y^2)));
Manipulate[ContourPlot[Im[((Complex[x,y]+(Sqrt[((Complex[x,y])^2)-4])-2Complex[Δx,Δy])/(2(Sqrt[(((1-Δx))^2)+((Δy)^2)])))+((2(Sqrt[(((1-Δx))^2)+((Δy)^2)]))/(Complex[x,y]+(Sqrt[((Complex[x,y])^2)-4])-2Complex[Δx,Δy]))],{x,0,3},{y,-1,1},ImageSize->Full],{Δx,-.2,-.05},{Δy,.025,.1},Deployed->True]