  A vector field on a K\"ahler manifold is called c-projective if its flow
preserves the J-planar curves. We give a complete local classification of
K\"ahler real 4-dimensional manifolds that admit an essential c-projective
vector field. An important technical step is a local description of
4-dimensional c-projectively equivalent metrics of arbitrary signature. As an
application of our results we prove the natural analog of the classical
Yano-Obata conjecture in the pseudo-Riemannian 4-dimensional case.
