In this paper, we develop a concise differential–potential framework for the functions of a generalized quaternionic variable in the two-parameter algebra
, with
. Starting from left/right difference quotients, we
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In this paper, we develop a concise differential–potential framework for the functions of a generalized quaternionic variable in the two-parameter algebra
, with
. Starting from left/right difference quotients, we derive complete Cauchy–Riemann (CR) systems and prove that, away from the null cone where the reduced norm
N vanishes, these first-order systems are necessary and, under
regularity, sufficient for left/right differentiability, thereby linking classical one-dimensional calculus to a genuinely four-dimensional setting. On the potential theoretic side, the Dirac factorization
shows that each real component of a differentiable mapping is
-harmonic, yielding a clean second-order theory that separates the elliptic (Hamiltonian) and split (coquaternionic) regimes via the principal symbol. In the classical case
, we present a Poisson-type representation solving a model Dirichlet problem on the unit ball
, recovering mean-value and maximum principles. For computation and symbolic verification, real
matrix models for left/right multiplication linearize the CR systems. Examples (polynomials, affine CR families, and split-signature contrasts) illustrate the theory, and the outlook highlights boundary integral formulations, Green kernel constructions, and discretization strategies for quaternionic PDEs.
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