  In this article, we investigate four-dimensional gradient shrinking Ricci
solitons close to a K\"ahler model. The first theorem could be considered as a
rigidity result for the K\"ahler-Ricci soliton structure on $\mathbb{S}^2\times
\mathbb{R}^2$ (in the sense of Remark 1). Moreover, we show that if the
quotient of norm of the self-dual Weyl tensor and scalar curvature is close to
that on a K\"ahler metric in a specific sense, then the gradient Ricci soliton
must be either half-conformally flat or locally K\"ahler.
