We explore distributional chaos for a \(C_0\) -semigroup of linear operators on a Banach space whose index set is a sector in the complex plane. We establish the relationship between distributional sensitivity and distributional chaos by characterizing them in terms of distributionally (semi-)irregular vectors. Additionally, we provide conditions under which a \(C_0\) -semigroup admits a linear manifold of distributionally irregular vectors. Furthermore, we delve into the study of distributional chaos for the translation \(C_0\) -semigroup on weighted \(L_p\) -spaces with a complex sector as the index set. We obtain a sufficient condition for dense distributional chaos, expressed in terms of the weight. In particular, we construct an example of a translation \(C_0\) -semigroup with a complex sector index set that is Devaney chaotic but not distributionally chaotic.
