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Is there an extensible formula for N-dimensional quaternions?

What would the formula look like for example for 2, 3, 4 and 5 functional dimensions of rotation (where 3 would be the normal quaternion?)

Examples of associated rotations and quaternion group cayley tables extending beyond the normal quaternion would also be very helpful.

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If, by "$N$-dimensional quaternions" you mean "an algebra that can model rotations of $N$-space," then the answer is that you need to read up on rotors in geometric/Clifford algebra algebra.

In dimensions higher than $3$, it becomes more difficult to describe (no matter what definition you adopt) what the rotations are, so I don't think "the formula" you are looking for exists, although it might for specialized cases.

For $n=2$ you do not even need geometric algebras and the rotor "sandwich" product: the complex numbers with modulus $1$ already handle rotation via multiplication.

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