An antilinear operator in complex vector spaces is an important operator in the study of modern quantum theory, quantum and semiclassical optics, quantum electronics and quantum chemistry. Consimilarity of complex matrices arises as a result of studying an antilinear operator referred to different bases in complex vector spaces, and the theory of consimilarity of complex matrices plays an important role in the study of quantum theory. This paper, by means of a real representation of a complex matrix, studies the relation between consimilarity and similarity of complex matrices, sets up an algebraic bridge between consimilarity and similarity and turns the theory of consimilarity into that of ordinary similarity. This paper also gives some applications of consimilarity of complex matrices.