The Zeta Axis: A Three-Dimensional Visualisation Convention for Arbitrary Complex Functions via the Field of Formal Laurent Series

Classical complex analysis introduces structural singularities, such as poles and branch cuts, which traditionally yield undefined values or geometric discontinuities at critical points. This paper presents the "Zeta Axis", an algebraic framework derived from intuitive geometric postulates that treat the classical zero not as an absolute void, but as an invertible unit \(z\) generating a third dimension. We demonstrate that formalizing these postulates leads to the well-established field of formal Laurent series \(\mathbb{C}((z))\), bridging intuitive infinitesimal arithmetic with non-standard analysis, Clifford's dual numbers, and forward-mode automatic differentiation. The primary contribution of this work is a novel three-dimensional visualization convention. By evaluating arbitrary analytic functions at a shifted input line \(w + z\), we map the classical output and its first-order jet (derivative) simultaneously into a single 3D space spanned by real, imaginary, and zeta axes. The resulting surfaces render algebraic identities as rigorous geometric structures—such as the exact hyperbolic paraboloid formed by \(f(x) = 1/x\)—while relocating poles into clean, structurally informative geometric boundaries. We outline the current MATLAB implementation, discuss inherent limitations regarding transcendental infinitesimal inputs, and propose future extensions including Puiseux series scaling and multivariate jet spaces.