Semilinear operators on a complex Hilbert space are studied in a part of a program that aims to develop the theories of additive operators on complex and quaternionic Hilbert spaces for application to problems in mathematical physics. The more notable among the new results proved on the eigenvalue problem for semilinear operators are the following: (i) if α is an eigenvalue of a semilinear operator then so also is any complex number which has the same modulus as α; (ii) if a normal semilinear operator has two eigenvectors belonging to different eigenvalues, then either the two eigenvectors are orthogonal or two eigenvalues have the same moduli; and (iii) a normal semilinear operator has a complete set of eigenvectors if and only if it is self-adjoint. Further, it is shown that there exists a norm-preserving semilinear isomorphism between the spaces of bounded linear and semilinear operators on a complex Hilbert space. Finally it is demonstrated how the theory of semilinear operators can be exploited to solve the problems of finding three involutive mutually anticommuting self-adjoint two-by-two matrices and four four-by-four matrices with the same properties: the unusual and remarkably easy solution of this old familiar exercise establishes the relevance of the theory being developed here to physics.