A fast algorithm, named Complex-Valued Fast Frobenius DIAGonalization (CVFFDIAG), is proposed for seeking the nonunitary approximate joint diagonalizer of a given set of complex-valued target matrices. It adopts a multiplicative update to minimize the Frobenius-norm formulation of the approximate joint diagonalization problem. At each of multiplicative iterations, a strictly diagonally dominant updated matrix is obtained. This scheme ensures the invertibility of the diagonalizer. The CVFFDIAG relaxes several constraints on the target matrices and thus has much general applications. Furthermore, the special approximation of the cost function, the ingenious utilization of some structures and the adequate notation of concerned variables lead to the high computational efficiency of the proposed algorithm. Numerical simulations are conducted to illustrate good performances of the CVFFDIAG.