After asking this question, I figured out that I am also interested in the following related question:

Is Abhyankar-Moh theorem 1.6 still valid if we remove the algebraic closedness assumption?

The answer is probably no, but I do not know how to find a counterexample over $\mathbb{R}$, for example.

Moreover, if I am not wrong, AM theorem 1.6 appears as Theorem 1 of van den Essen's paper; I do not see where algebraic closedness is needed in the proof.

Thank you very much for any comments or hints!

**Edit:** I have recently found this beautiful criterion (Theorem 3.3), which does not require algebraic closedness of the base field.