It is well-known that if one assumes algebraic closedness and characteristic 0 of the residue field then finite covers of complete DVRs are all of the form $A[x]/(x^m-a)$ for some $a \in A$ (direct sums of such, more precisely).
Is there a similar description for finite covers of complete regular local rings in higher dimensions?
Depending on what you mean by cover your statement isn't true in the DVR case. To make it true in that case you can throw in the condition that the cover be normal and it may also be a good idea to assume the map is flat. However, under your assumptions the complete regular local ring is a power series ring over the residue field $k$ in $n$ say variables. Now, for $n$-dimensional complete local normal ring $R$ with residue field $k$ there is a finite map from $R$ to the power series ring in $n$ variables. If $R$ is also Cohen-Macaulay then the map is flat. Hence, any $n$-dimensional complete local normal Cohen-Macaulay ring with $k$ as residue field appears as covering. When $n>1$ there is an enormous number of possible such ring so those of the suggested type only cover a very small number of cases (for instance they have embedding dimension at most $n+1$ and there are normal Cohen-Macaulay local rings of fixed dimension and arbitrary high embedding dimension).
