  Two proper polynomial maps $f_1, f_2 \colon \mathbb{C}^2 \longrightarrow
  \mathbb{C}^2$ are said to be \emph{equivalent} if there exist $\Phi_1, \Phi_2
\in \textrm{Aut}(\mathbb{C}^2)$ such that $f_2=\Phi_2 \circ f_1 \circ \Phi_1$.
  We investigate proper polynomial maps of arbitrary topological degree $d \geq
2$ up to equivalence. Under the further assumption that the maps are Galois
coverings we also provide the complete description of equivalence classes. This
widely extends previous results obtained by Lamy in the case $d=2$.
