[Abhyankar-Moh theorem][1] 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? 

Also, it was mentioned in [Wikipedia][2] 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!


  [1]: https://eudml.org/doc/151610
  [2]: https://en.wikipedia.org/wiki/Abhyankar%E2%80%93Moh_theorem