In this paper, we investigate the geometry of 4-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature (half PIC) or half nonnegative isotropic curvature. Our first main result is a certain quadratic curvature lower bound estimate for such gradient Ricci shrinkers. As a consequence, we obtain a new proof of the classification result first shown by Li-Ni-Wang [31] for gradient shrinking Kähler-Ricci solitons of complex dimension two with nonnegative isotropic curvature. Moreover, we classify 4-dimensioinal complete gradient shrinking Ricci solitons with half PIC under an additional assumption that their Ricci tensor has an eigenvalue with multiplicity 3.