Consider a polynomial with degree n. Then the greatest number of stationary points it may have is n-1. How can we build intuition or prove for why this is the case?
However, my main question is as follows. Consider a polynomial of degree 4. It is possible for that polynomial to have 1 stationary point. Where do the other stationary points go? Could it be they overlap each other and count as a single stationary point?
The fundamental theorem of algebra (FTA) states: every non-zero, single-variable, degree $n$ polynomial with complex coefficients has counted with multiplicity, exactly $n$ complex roots.
If $P(x)$ is a polynomial of degree $n$, it is clear by the power rule that its derivative $P'(x)$ will have degree $n-1$. Since the stationary points of $P(x)$ are defined to occur at the roots of $P'(x)$, FTA tells us that $P(x)$ will have at most $n-1$ stationary points.
As for your question, FTA only states that $P'(x)$ will have exactly $n-1$ complex roots. It may just so happen that some of those roots are real. Those real roots will correspond to stationary points while the remaining non-real ones correspond to the "missing" stationary points.