You're right. Quine proved in "On the self-intersections of the image of the unit circle under a polynomial mapping" that if the degree is $n$ and $f(z)\neq g(z^k)$ with $k>1$, then the number of points with at least 2 distinct preimage points is at most $(n-1)^2$. An example shows that this is sharp. Here's the review in MR.

You're right. Quine proved in "On the self-intersections of the image of the unit circle under a polynomial mapping" that if the degree is $n$ and $f(z)\neq g(z^k)$ with $k>1$, then the number of points with at least 2 distinct preimage points is at most $(n-1)^2$. An example shows that this is sharp. Here's the review in MR.

After the proof, there is a remark:

As a simple consequence of this theorem we note that a polynomial $p$ cannot map $|z| < 1$ conformally onto a domain with a slit, for in this case $p(e^{i\phi})$ would have an infinite number of vertices.