<< Graphics`Colors`

for[z_, a_] = (4  z^3 - 4)/(z^4 - 4  z + a)

Element[x, Reals]

Element[y, Reals]

z[x_, y_] = x + I y

f[x_, y_, a_] = for[z[x, y], a]

lev[x_, y_, a_] = Log[z[x, y]^4 - 4  z[x, y] + a]

level[x_, y_, a_] = Log[Abs[z[x, y]^4 - 4  z[x, y] + a]]

gr[b_] := 
        ContourPlot[(Abs[f[x, y, 0.5]]), {x, -4, 4}, {y, -4, 4}, 
          ContourShading -> False, PlotPoints -> 250, 
          ContourStyle -> {AbsoluteThickness[2], Blue}, 
          Contours -> {b}];

gr[2];

gr[1.688];

gr[1.683];

gr[1.68];

gr[1.65];

gr[1.61];

gr[1.605];

gr[1.6];

gr[1.51529];

gr[1.48];

gr[1.4];

gr[1.0];

gr[ .2];

gr[10];

Show[%22, %25]

Expand[f[x, y,  .5]]

Together[%]

u[x_, 
        y_] = (4 (-1 + x^3 + 3  I x^2  y - 3  x y^2 - 
                I y^3))/((1/2) - 4  x + x^4 - 4  I y + 
            4  I x^3  y - 6  x^2  y^2 - 4  I x y^3 + y^4)

ubar[x_, 
        y_] = (4 (-1 + x^3 - 3  I x^2  y - 3  x y^2 + 
                I y^3))/((1/2) - 4  x + x^4 + 4  I y - 
            4  I x^3  y - 6  x^2  y^2 + 4  I x y^3 + y^4)

w[x_, y_] = Simplify[u[x, y] ubar[x, y]]

w

w1[x_, y_] = D[w, x]

w[x, y]

Derivative[1][w[x, y]]

Clear[w]

w[x_, y_] = Simplify[u[x, y] ubar[x, y]]

w[t, u]

w1[x_, y_] = Simplify[Together[D[w[x, y], x]]]

w2[x_, y_] = Simplify[Together[D[w[x, y], y]]]

NSolve[{w1[x, y] == 0, w2[x, y] == 0}, {x, y}]

x1 = 0.7892636726169011; y1 = -0.8025375775293005;

x2 = 0.47233441354860933; y2 = -1.3762428078135358;

x3 = -1.5785273452338204; y3 = 0;

x4 -> -0.5000000000000256; y4 = 0.8660254037844072;

x5 = 0.9999999999999927; y5 = 0. ;

x6 = 0.4723344135491; y6 = 1.376242807814189;

x7 = -0.5000000000000293; y7 = -0.8660254037844842;

x8 = 0.7892636726171907; y8 = 0.8025375775296735;