<< Graphics`

vsign[z_] := {Re[Sign[z]], Im[Sign[z]]}

q[z_, e_] := e Cos[z]

N[vsign[Sqrt[q[2 + 2  I, 1]]]]

?vsign

PlotVectorField[
      vsign[Sqrt[q[x + I y, 1]]], {x, -2 Pi, 2  Pi}, {y, -2 Pi, 
        2  Pi}, PlotPoints -> 17]

stokes[{x_, y_}] = {y, x}

antistokes[{x_, y_}] = {x, -y}

PlotVectorField[
      stokes[vsign[Sqrt[q[x + I y, 1]]]], {x, -2 Pi, 
        2  Pi}, {y, -2 Pi, 2  Pi}, PlotPoints -> 33]

PlotVectorField[
      antistokes[vsign[Sqrt[q[x + I y, 1]]]], {x, -2 Pi, 
        2  Pi}, {y, -2 Pi, 2  Pi}, PlotPoints -> 33]