The biquaternion (BQ) algebra, realized at the Pauli-spin level as the full matrix algebra (M(2,mathbb{C})), provides a compact framework in which classical field theories can be organized through a common hierarchy of algebraic products and channel decompositions. This paper develops that hierarchy for both electrodynamics and relativistic fluid dynamics and explores the structural relationships that emerge between them.In electrodynamics, the BQ hierarchy reproduces the electromagnetic field, Lorentz force law, Maxwell equations, wave equations, and interaction-conservation relations from a sequence of algebraic products involving the four-current and four-potential. In relativistic fluid dynamics, the corresponding hierarchy generates the continuity equation, Bernoulli relation, vorticity transport, Euler—Lamb momentum balance, acoustic wave equations, helicity relations, and stress-energy conservation through analogous products of the fluid four-velocity and momentum-density field.The central result is that both theories share the same four-level algebraic structure and the same geometric channel decomposition. The force equations (JB=F) and (UH=F) emerge as the unique four-channel level of their respective hierarchies, revealing a common organization that is largely hidden in conventional formulations. The analysis further identifies the acoustic refractive index (n_s=c/c_s) as a hierarchy-wide parameter linking relativistic and practical fluid dynamics, and locates the greater complexity of fluid dynamics in the self-referential coupling between velocity and momentum-density fields.The paper does not introduce new physical postulates. Rather, it proposes that the BQ channel decomposition provides a unified organizational framework that makes structural relationships between electrodynamics and fluid dynamics explicit, offering a common language for force, energy, momentum, vorticity, helicity, and wave phenomena across both theories. In this sense, the work is intended as a conceptual and educational framework as much as a mathematical formalism, and as a foundation for future applications in gravitation, magnetohydrodynamics, and AI-assisted symbolic physics.