Motivated by the fact that there exists the operation of conjugation in quantum systems, the concept of bicon-numbers is proposed in this article. The bicon-numbers are defined by introducing two symbolic parameters into the set of complex numbers. The basic functions of these two symbolic parameters are specified by an axiom which abstracts the operation of complex conjugation. Basic properties are developed for the operations of addition and multiplication in the bicon-number set. In addition, several different forms are given for bicon-numbers, and the corresponding operation rules are established. By exploring the relations of the vensors in the bicon-number set, the structure of the bicon-number set is depicted, and real matrix representations of bicon-numbers are also presented. Besides, bicomplex matrix representations for bicon-numbers are also investigated in view that the operation of multiplication for bicomplex numbers possesses commutativity property. In addition, the matrices with bicon-numbers as entries are investigated, and state responses of some quantum systems are given within the framework of bicon-numbers.