In this paper, we study the asymptotic behavior of the outliers of the sum a Hermitian random matrix and a finite rank matrix which is not necessarily Hermitian. We observe several possible convergence rates and outliers locating around their limits at the vertices of regular polygons as in a previous work by Benaych-Georges and Rochet, as well as possible correlations between outliers at macroscopic distance as in works by Knowles, Yin, Benaych-Georges and Rochet. We also observe that a single spike can generate several outliers in the spectrum of the deformed model, as already noticed in several previous works. In the particular case where the perturbation matrix is Hermitian, our results complete a previous work of Benaych-Georges, Guionnet and Maïda, as we consider fluctuations of outliers lying in "holes" of the limit support, which happen to exhibit surprising correlations.