You're right.  Quine proved in "[On the self-intersections of the image of the unit circle under a polynomial mapping][1]" 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][2] 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.



  [1]: http://www.jstor.org/stable/2039005
  [2]: http://www.ams.org/mathscinet-getitem?mr=313485