Foundational aspects of a new matrix holomorphic structure

Purpose In this paper we talk about complex matrix quaternions (biquaternions) and we deal with some abstract methods in mathematical complex matrix analysis. Design/methodology/approach We introduce and investigate the complex space HC consisting of all 2 × 2 complex matrices of the form ξ=z1+iw1z2+iw2−z‾2−iw‾2z‾1+iw‾1, (z1,w1,z2,w2)∈C4 . Findings We develop on HC a new matrix holomorphic structure for which we provide the fundamental operational calculus properties. Originality/value We give sufficient and necessary conditions in terms of Cauchy–Riemann type quaternionic differential equations for holomorphicity of a function of one complex matrix variable ξ∈HC . In particular, we show that we have a lot of holomorphic functions of one matrix quaternion variable.