Let $K$ be a field. Say that *polynomials are almost surjective* over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. That is, for all but finitely many $a\in K$, $f(x)-a$ has a root.

Clearly polynomials are almost surjective over any finite field, or over any algebraically closed field. My question is whether the converse holds. That is:

If $K$ is an infinite field and polynomials are almost surjective over $K$, must $K$ be algebraically closed?

(This answer to a similar question gave a simple proof that $\mathbb{C}$ is algebraically closed from the fact that polynomials are almost surjective over $\mathbb{C}$. However, this proof made heavy use of special properties of $\mathbb{C}$ such as its topology, so it does not generalize to arbitrary fields.)