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We first define a new Gray map from $R=\mathbb{Z}_4+u\mathbb{Z}_4$ to$\mathbb{Z}^{2}_{4}$, where $u^2=1$ and study $(1+2u)$-constacyclic codes over$R$. Also of interest are some properties of $(1+2u)$-constacyclic codes over$R$. Considering their $\mathbb{Z}_4$ images, we prove that the Gray images of$(1+2u)$-constacyclic codes of length $n$ over $R$ are cyclic codes of length$2n$ over $\mathbb{Z}_4$. In many cases the latter codes have better parametersthan those in the online database of Aydin and Asamov. We also give a correctedversion of a table of new cyclic $R$-codes published by \"Ozen et al. in FiniteFields and Their Applications, {\bf 38}, (2016) 27-39.

On constacyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$ and their Gray images