  Quaternion (Q-) mathematics formally contains many fragments of physical
laws; in particular, the Hamiltonian for the Pauli equation automatically
emerges in a space with Q-metric. The eigenfunction method shows that any
Q-unit has an interior structure consisting of spinor functions; this helps us
to represent any complex number in an orthogonal form associated with a novel
geometric image (the conic-gearing picture). Fundamental Q-unit-spinor
relations are found, revealing the geometric meaning of spinors as Lam\'e
coefficients (dyads) locally coupling the base and tangent surfaces.
