Aydogdu et al. studied the standard forms of generator and parity-check matrices of $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive codes, and presented generators of $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive cyclic codes (Finite Fields Appl. 48: 241-260, 2017). In this paper, we investigate some other useful properties of $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive codes, including asymptotically good $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive cyclic codes and $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive complementary dual codes. The present paper can be viewed as a necessary complementary part of Aydogdu’s work.