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The classical Chern correspondence states that a choice of Hermitian metric
on a holomorphic vector bundle determines uniquely a unitary 'Chern
connection'. This basic principle in Hermitian geometry, later generalized to
the theory of holomorphic principal bundles, provides one of the most
fundamental ingredients in modern gauge theory, via its applications to the
Donaldson-Uhlenbeck-Yau Theorem. In this work we study a generalization of the
Chern correspondence in the context of higher gauge theory, where the structure
group of the bundle is categorified. For this, we define connective structures
on a multiplicative gerbe and propose a natural notion of complexification for
an important class of 2-groups. Using this, we put forward a new notion of
higher connection which is well-suited for describing holomorphic principal
2-bundles for these 2-groups, and establish a Chern correspondence in this way.
As an upshot of our construction, we unify two previous notions of higher
connections in the literature, namely those of adjusted connections and of
trivializations of Chern-Simons 2-gerbes with connection.