Abhyankar-Moh theorem says that if $L$ is a complex line in the complex affine plane $\mathbb{C}^2$, then every embedding of $L$ into $\mathbb{C}^2$ extends to an automorphism of the plane. It seems that one can replace $\mathbb{C}$ by any algebraically closed field $k$ of characteristic zero.

Are there 'similar' results for fields other than $k$, especially fields of characteristic zero, not necessarily algebraically closed?

(Probably there are easy counterexamples to the above claim, but maybe something similar can still be said?).

Also, it was mentioned in Wikipedia that there are generalizations of this theorem to higher dimensions.

Can one please explain what those generalizations are or give a reference?

Any comments are welcome!