I'm struggling with the definition of the stream function. Please check if my following understanding correct:

First we start with an undefined analytical function that we are going to call "complex potential": $W=\phi+i\psi$. From the properties of analytic functions, we have $\frac{dW}{dz}=\phi_x-i\phi_y$. Now, we are going to define $\phi$: $\nabla{\phi}=(\phi_x,\phi_y):=(u,v)$, where $(u,v)$ is the velocity field of our given 2-dimensional flow. We call $\phi$ the velocity potential function. Now, we are going to define the stream function as the imaginary part $\psi$ of the complex potential $W$. The uniqueness follows from analyticity of $W$.

We can now obtain the physical meaning of $\psi$ by noticing that from the Cauchy-Riemann equations it follows that $(\phi_x,\phi_y)=(\psi_y,-\psi_x)$. Since $(\psi_x,\psi_y)\cdot(\psi_y,-\psi_x)=0$, tangent curves of $\phi$ are orthogonal to the tangent curves of $\psi$. Now i refer to the theorem that states: "Theorem: If the function f is differentiable, the gradient of f at a point is either zero, or perpendicular to the level set of f at that point.". From this theorem and $\nabla{\phi}=(\phi_x,\phi_y):=(u,v)$ it follows that level curves of $\phi$ are orthogonal to the velocity field $(u,v)$. And since tangent curves of $\phi$ are orthogonal to the tangent curves of $\psi$, i conclude that level curves of $\psi$ are parallel to the velocity field $(u,v)$, in other words, the fluid flow follows the lines $\psi=const$.

Is this sound or not? When i started explaining the stream function like i wrote above, the professor said that my question was about the stream function and not about the complex potential. And my answer wasn't accepted.