Summary: This paper studies \(\lambda\)-constacyclic codes and skew \(\lambda \)-constacyclic codes over the finite commutative non-chain ring \(R = \mathbb{Z}_4 + u\mathbb{Z}_4 + u^2\mathbb{Z}_4\) with \(u^3 = 0\) for \(\lambda = (1 + 2u + 2u^2)\) and \((3 + 2u + 2u^2)\). We introduce distinct Gray maps and show that the Gray images of \(\lambda\)-constacyclic codes are cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes over \(\mathbb{Z}_4\). It is also shown that the Gray images of skew \(\lambda\)-constacyclic codes are quasi-cyclic codes of length \(2n\) and index 2 over \(\mathbb{Z}_4\). Moreover, the structure of \(\lambda\)-constacyclic codes of odd length \(n\) over the ring \(R\) is determined and give some suitable examples.