Summary: We mainly develop the foundation of a new function theory of several complex variables with values in a complex Clifford algebra defined on some subdomains of \(\mathbb{C}^{n+1}\), so-called complex holomorphic Cliffordian functions. We define the complex holomorphic Cliffordian functions, study polynomial and singular solutions of the equation \(D\Delta^mf=0\), obtain the integral representation formula for the complex holomorphic Cliffordian functions with values in a complex Clifford algebra defined on some submanifolds of \(\mathbb{C}^{n+1}\), deduce the Taylor expansion and the Laurent expansion for them and prove an invariance under an action of a Lie group for them.