In this paper, we introduce the Hermitian completely positive (HCP) matrix, which has a Hermitian completely positive (HCP) decomposition with all real and imaginary parts of the decomposition vectors being nonnegative. Some properties of the Hermitian completely positive matrix are given. A semidefinite algorithm is also proposed for checking whether a Hermitian matrix is HCP or not. If a matrix is not HCP, a certificate for it can be obtained; if it is, an HCP decomposition can be obtained.