Identifiability analysis of complex Blind Source Separation (BSS), ie to study under what conditions the BSS problem can be solved, is a longstanding and most critical problem in the community. It serves not only as the indicator to solvability of the BSS problem, but also as the constructive ground for developing efficient algorithms. Various BSS methods are based on jointly diagonalizing a set of matrices, which are generated using second-or higher-order statistics. The present work provides a general result on the uniqueness conditions of matrix joint diagonalization. It unifies all existing results on the identifiability conditions of complex BSS, with respect to non-circularity, non-stationarity, non-whiteness, and non-Gaussianity. Additionally, following the main identifiability result, a solution for complex BSS is proposed. It is given in closed form in terms of an eigenvalue and a singular value decomposition of two matrices.