The High Church of Quaternions comprises a set of mathematical, scientific, and philosophical ideas stemming from the mathematical system known as "quaternions", a 4-dimensional hypercomplex number system discovered by William Rowan Hamilton in 1843. The Church's name was mockingly coined by Eric Temple Bell in the brilliant but opinionated book Men of Mathematics, in the chapter entitled An Irish Tragedy.

The cross product of Quaternions can be right-handed or left-handed; that of Octonions more complicated ("untwisted" or "twisted"). The octonions are comprised of 7 overlapping quaternionic triads, each of which can have a right- or left-handed crossproduct. If certain interrelationships among the handednesses of these triads are not observed, one obtains the Twisted Octonions, rather than the usual Octonions. The Twisted Octonions have a natural decomposition into a 4-vector and a spinor, suggesting a relation to superspace and supersymmetry.

Perhaps there is also a relationship to the body-mind and matter-spirit dualities. We understand the Universe as an assemblage of material objects in 4-dimensional Space-Time. If Space-Time- is actually 8-dimensional, with extra "spin" coordinates, then all phenomena have extra degrees of freedom, and the biochemical phenomena of the brain, as well as the self-organizing principle powering biological evolution, might have geometrical structure in the extra dimensions that causes Consciousness to permeate all things.