Linear complementary pairs (LCP) of codes play an important role in armoring implementations against side-channel attacks and fault injection attacks. One of the most common ways to construct LCP of codes is to use Euclidean linear complementary dual (LCD) codes. In this paper, we first introduce the concept of linear codes with $σ$ complementary dual ($σ$-LCD), which includes known Euclidean LCD codes, Hermitian LCD codes, and Galois LCD codes. As Euclidean LCD codes, $σ$-LCD codes can also be used to construct LCP of codes. We show that, for $q > 2$, all q-ary linear codes are $σ$-LCD and that, for every binary linear code $\mathcal C$, the code $\{0\}\times \mathcal C$ is $σ$-LCD. Further, we study deeply $σ$-LCD generalized quasi-cyclic (GQC) codes. In particular, we provide characterizations of $σ$-LCD GQC codes, self-orthogonal GQC codes and self-dual GQC codes, respectively. Moreover, we provide constructions of asymptotically good $σ$-LCD GQC codes. Finally, we focus on $σ$-LCD Abelian codes and prove that all Abelian codes in a semi-simple group algebra are $σ$-LCD. The results derived in this paper extend those on the classical LCD codes and show that $σ$-LCD codes allow the construction of LCP of codes more easily and with more flexibility.