How can I find solution and plot multi complex variable equation?

Question

I have an question about solving equation.

x and y are variable, and a, b, and c are complex constant.

The equations is formed like: $x^2/a - y^2/b = c$.

I want to contour plot a graph range from $x1 < Re(x) < x2$ to $y1 < Re(y) < y2$. $x1$, $x2$, $y1$, and $y2$ is real number even though $x$ and $y$ satisfied this equation are complex number.

I have used

ContourPlot[x^2/a - y^2/b == c, {x, x1, x2}, {y, y1, y2}]

However, because this considers only real case, I can't find a solution.

If anyone has a idea, please help me.

Thank you.

Answer 1

Something like this using ReIm?

SeedRandom[1]

{a, b, c} = RandomComplex[1 + I, 3];

ContourPlot[ReIm[x^2/a - y^2/b] == ReIm[c], {x, -1, 1}, {y, -1, 1}]

If you are using an older version of Mathematica without ReIm define:

ReIm = Function[x, {Re@x, Im@x}, Listable];