We consider \(N\times N\) matrices with complex entries that are perturbed by a complex Gaussian matrix with small variance. We prove that if the unperturbed matrix satisfies certain local laws then the bulk correlation functions are universal in the large N limit. Assuming the entries are independent and identically distributed (iid) with a common distribution that has finite moments, the Gaussian component is removed by the four moment theorem of Tao and Vu.
