Complex numbers can describe 2d rotations. For 3d rotations, we need 2 more imaginary units and not just 1 more because though there's just 1 more dimension, there's 2 more 2d subspaces. Adding a dimension to an n-d space creates n more pairs of basis vectors, not 1 more pair.
There are rotations in 2d, 4d and 8d. In 4d, you lose commutativity, in 8d associativity. The key for the relationship between geometry and numbers are Clifford algebras
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Do we know that there is no Clifford algebra structure inside the sedenions?
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