<< Graphics`PlotField`

example 9.3   p.212

ContourPlot

fz = z^2; 
    w = Expand[fz /. z ->  x +  I y]
{ϕ = ComplexExpand[Re[w]], ψ = 
        ComplexExpand[Im[w]]}

{D[ϕ, x] - D[ψ, y], 
      D[ϕ, y] + D[ψ, x]}

zu11 = 
      ContourPlot[ψ, {x, 0, 3}, {y, 0, 3}, 
        AspectRatio -> Automatic, ContourShading -> False, 
        Contours ->  20]

zu12 = 
      ContourPlot[ϕ, {x, 0, 3}, {y, 0, 3}, 
        AspectRatio -> Automatic, ContourShading -> False, 
        Contours ->  10]

aaa = Show[zu11, zu12]

zu13 = 
      PlotGradientField[ϕ, {x, 0, 3, 0.4}, {y, 0, 3, 0.25}, 
        MaxArrowLength ->  10]

Show[zu11, zu13]

NDSolve  Ex 9.3

eq = z'[t] == 
        Conjugate[D[fz, z] /. z -> z[t]]

ne = 10;
z0 = Table[30 I  + 0.01 i, {i, ne}];

zuall = 
      Table[{ans = NDSolve[{eq, z[0] == z0[[i]]}, 
                z[t], {t, 0, 10}][[1]]; 
          ParametricPlot[{Re[z[t]] /. ans, Im[z[t]] /. ans}, {t, 0, 10}, 
            PlotRange ->  {{0, 3}, {0, 3}}, 
            DisplayFunction ->  Identity]}, {i, ne}]

zu14 = 
      Show[zuall, DisplayFunction ->  $DisplayFunction]

Show[zu14, zu13]