The term "hypercomplex" was formulated to denote quaternions, octonions and extensions beyond the complex number system. But it is also used to include the complex, applying it to generate the rest.

• The 1-hypercomplex has the polynomial form: h₁ = r₁ + r₂i, where the r_i are real numbers, and i is a new unit such that i² = ¯1.
• The 2-hypercomplex (quaternion) has the polynomial form: h₂ = c₁ + c₂j, where j is a new unit (j ¹ i), such that j² = ¯1. From the polynomial forms, c₁ = r₁ + r₂i and c₂ = r₃ + r₄i, we find, by subsitution: h₂ = c₁ + c₂j = r₁ + r₂i + r₃j + r₄ij = r₁ + r₂i + r₃j + r₄k, for ij º k, and ji = ¯k, with a another new unit, k ¹ i, k ¹ j, and k² = ¯1.
• The 3-hypercomplex (octonion) has the polyonomial form: h₃ = q₁ + q₂l, with a new unit, l ¹ i, l ¹ j, l ¹ k, and l² = ¯1. From the polynomial forms, q₁ = r₁ + r₂i + r₃j + r₄k and q₂ = r₅ + r₆i + r₇j + r₈k, we have by substitution: h₃ = r₁ + r₂i + r₃j + r₄k + r₅l + r₆li + r₇lj + r₈lk = r₁ + r₂i + r₃j + r₄k + r₅l + r₆m + r₇n + r₈o, for new units li º m, lj º = n, lk º = o, such that all of the UNITS are distinct but have squares equal to ¯1.
• The 4-hypercomplex can be generated from 3-hypercomplex numbers, etc. In general, the (n + 1)-hypercomplex can be generated from n-hypercomplex numbers.

(Note: George Temple, in 100 Years of Mathematics, says that Hamilton labeled "quaternions, after the Vulgate name for the squads of four soldiers who guarded St. Petern prison (Acts19:4)".)