Let C be a conjugation on a complex separable Hilbert space H. A bounded linear operator T is said to be C-normal if (Formula presented.). In this paper, first, we give a representation of C-normal operators on finite dimensional Hilbert space and later extend it to compact C-normal operators on infinite-dimensional separable Hilbert spaces. In the end, we discuss the eigenvalue problem for C-normal operators and show that every compact C-normal operator has a solution for the eigenvalue problem. © 2022 Informa UK Limited, trading as Taylor & Francis Group.