Consider a random matrix of size N as an additive deformation of the complex Ginibre ensemble under a deterministic matrix X 0 with a finite rank, independent of N . When some eigenvalues of X 0 separate from the unit disk, outlier eigenvalues may appear asymptotically in the same locations, and their fluctuations exhibit surprising phenomena that highly depend on the Jordan canonical form of X 0 . These findings are largely due to Benaych-Georges and Rochet [9], Bordenave and Capitaine [11], and Tao [49]. When all eigenvalues of X 0 lie inside the unit disk, we prove that local eigenvalue statistics at the spectral edge form a new class of determinantal point processes, for which correlation kernels are characterized in terms of the repeated erfc integrals. This thus completes a non-Hermitian analogue of the BBP phase transition in Random Matrix Theory. Similar results hold for the deformed quaternion Ginibre ensemble.