The general question that one can ask is if embeddings $\iota:\mathbb{A}^k\hookrightarrow\mathbb{A}^n$ can be rectified, i.e., if there is an automorphism of $\mathbb{A}^n$ which takes $\iota$ to the standard embedding of a coordinate $k$-plane. The Abhyankar-Moh(-Suzuki) theorem is the case where $k=1$ and $n=2$. The case where $k=n-1$ is the Abhyankar-Sathaye conjecture.

There are several things that could be called higher-dimensional generalization. For example, there are results of Craighero, Jelonek, Nori, Srinivas, etc... which show that embeddings between affine spaces are rectifiable if the codimension is high enough. Here are some references:

P. Craighero. A result on $m$-flats in $\mathbb{A}^n_k$. Rend. Sem. Math. Univ. Padova 75 (1986), 39-46.

Z. Jelonek. The extension of regular and rational embeddings. Math. Ann. 277 (1987), 113-120. (link to paper on Springer website)

V. Srinivas. On the embedding dimension of an affine variety. Math. Ann. 289 (1991), 125-132. (link to paper on Springer website)

The most general result, proved as Theorem 2 in the paper of Srinivas above (and attributed to Nori) states that an embedding $\mathbb{A}^k$ in $\mathbb{A}^n$ is rectifiable if $n\geq 2k+2$. Note that the base field for Srinivas' theorem can be any infinite field.

Embeddings of other subvarieties could also be considered:

- S. Kaliman. Extensions of isomorphisms between affine algebraic subvarieties of $k^n$ to automorphisms of $k^n$. Proc. Amer. Math. Soc. 113 (1991), 325–334. (link to paper on AMS website)

There is also survey of van den Essen with some more literature references on generalizations of Abhyankar-Moh and related questions:

- A. van den Essen. Around the Abhyankar-Moh theorem. In: Algebra, arithmetic and geometry with applications (Papers from Shreeram S. Abhyankar's 70th birthday conference), Springer (2004), pp. 283-294. (link to paper on Springer website)