An old theorem going back to Boothby states that a compact Hermitian manifold with Chern curvature vanishing identically is a compact quotient of a complex Lie group with a left invariant metric. Are there analogous classification results in the complete non-compact setting? This question is perhaps too broad -- we might have to restrict attention to complete non-compact Hermitian manifolds with prescribed volume growth.
Update: It was shown by Chen--Chen--Ni (see Example 3.9 in the arXiv reference below) that there are (complete) Hermitian metrics on $\mathbb{C}^n$ with vanishing holomorphic sectional curvature with non-vanishing curvature: Take $\omega = \sqrt{-1} \partial \overline{\partial} \log (1+ | z |^2)$ and $f = 2 \log(1 + | z |^2)$. Then $\widetilde{\omega} = e^f \omega$ is a Hermitian metric (conformally Kähler) with vanishing holomorphic sectional curvature and non-vanishing curvature.
Chen, H., Chen, L., Ni, L., CHERN-RICCI CURVATURES, HOLOMORPHIC SECTIONAL CURVATURE AND HERMITIAN METRICS, arXiv: https://arxiv.org/pdf/1905.02950.pdf
