The purpose of this paper is to get new generating relations of the complex Hermite polynomials $H_{p,q}\left ( z,z^{*} \right )$ by extending the realization $\uparrow_{\omega,\mu}$ to study multiplier representations of a Lie group G(0,1). Also, we establish new operational formulas involving the polynomials $H_{p,q}\left ( z,z^{*} \right )$ and we will use them in a simple way to obtain new generating functions for the polynomials $H_{p,q}\left ( z,z^{*} \right )$. Also, we derive some special cases which are worth interest.