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.
Is it possible to replace $\mathbb{C}$ by $k$, where $k$ is: (1) any algebraically closed field of characteristic zero? I guess that the answer is yes. (2) any field of characteristic zero? Perhaps there exists an easy counterexample with $k=\mathbb{R}$. (3) any (commutative) integral domain of characteristic zero?
Any comments are welcome!
Edit: (1) This explanation about Abhyankar-Moh results is relevant.
(2) From the discussion in the comments, my first question has a positive answer due to Theorem 1.6, and my second question should have a counterexample over $\mathbb{R}$. The comments are good, since they clarify my question.