What you want to achieve is a visualization of the complex function f(z) = u(z) + i v(z) with z = x + i y.
In your image, you see two contour plots in the same axis: one for u(z) and one for v(z).
There are only two questions remaining:
how can you plot two contour plots of given functions into the same axis, and how can you control their appearance?
what are the formulas for u(z) and v(z)?
I can assist you with 1. Concerning 2., I have not been able to reproduce you graph by means of your provided functions. Can you verify them? Are you sure that u(z) = u(x) does not depend on y? And, similarly, that v(z) = v(y) does not depend on x? You image seems to indicate that u and v depend on both x and y. And what is alpha? It seems to be something like pi/5 (that's what I guessed).
So, here is a solution for question (1). Suppose we want to visualize f(z) = z^2 = (x+iy)^2 = (x^2 - y^2) + xy i :
\documentclass{article}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
view={0}{90},
xlabel=real axis,
ylabel=imaginary axis,
]
\addplot3[contour gnuplot={number=9,labels=false,draw color=blue}] {x^2-y^2};
\addplot3[contour gnuplot={number=9,labels=false,draw color=red}] {x*y};
\end{axis}
\end{tikzpicture}
\end{document}

Here, I used draw color
to change the color (which is typically mapped color
for contour plots).