c=1-2I

f[z_]:=c z

ComplexExpand[f[x+I y]]

phi[x_,y_]:=ComplexExpand[Re[f[x+I y]]]

psi[x_,y_]:=ComplexExpand[Im[f[x+I y]]]

phi[x,y]

psi[x,y]

g1=ContourPlot[phi[x,y]==Table[c,{c,-5,5,1.0}],{x,-2,2},{y,-2,2},ContourStyle->Directive[Blue,Thin]]

g2=ContourPlot[psi[x,y]==Table[c,{c,-5,5,1.0}],{x,-2,2},{y,-2,2},ContourStyle->Directive[Red,Thin]]

Show[g1,g2]

vv[x_,y_]:={Re[f'[x+I y]],-Im[f'[x+I y]]}

g3=VectorPlot[vv[x,y],{x,-2,2},{y,-2,2}]

Show[g1,g2,g3]