Plotting level curves for orthogonal analytic functions u, and v

Questions
,
1

I’m the Wolfram-U Complex Analysis lecture series. In the example 3 of the section on Harmonic functions we are asked to plot some level curves for the real and imaginary parts of a complex function.
Given f(z) = z^2
I’m trying to plot the level curves for

u[x_,y_]:= x^2-y^2

and

v[x_,y_]:= 2*x*y

I have tried ContourPlot and ComplexContourPlot what am I missing?

2

Have you tried

ContourPlot[{x^2 - y^2, 2 x y}, {x, -2, 2}, {y, -2, 2}, 
 ContourShading -> False]
ComplexContourPlot[ReIm[z^2], {z, 2}]
3

That worked like a treat. Thank you!