Consider the function $f(z)=z^2$. Prove that level curves of $Re(f(z))$ and $Im(f(z))$ at $z=1+2i$ are orthogonal to each other.
I am not sure how to apply level curves or contour lines for complex variables. As far as real variables go, I am aware that for a function like $f(x,y) = \sqrt{x^2+y^2}$, the level curves are the circles centered at $(0,0)$. For this question would it be sufficient for me to take the gradients of the real components and the imaginary components and show that its dot product equals to $0$? But if I do that, how am I showing that the level curves are orthogonal to each other?