  This work presents an extension of the Construction $\pi_A$ lattices proposed
in \cite{huang2017construction}, to Hurwitz quaternion integers. This
construction is provided by using an isomorphism from a version of the Chinese
remainder theorem applied to maximal orders in contrast to natural orders in
prior works. Exploiting this map, we analyze the performance of the resulting
multilevel lattice codes, highlight via computer simulations their notably
reduced computational complexity provided by the multistage decoding. Moreover
it is shown that this construction effectively attain the Poltyrev-limit.
