If we start with a hypercomplex number with $2^n$ entries, how quickly can we multiply it by another hypercomplex number, modulo a prime?
EXAMPLE
For example, with $n=1$, we get the complex numbers. So we are multiplying a number, say $a=a_r + a_i i$ by $b=b_r + b_i i$ which equals: $$(a_r\cdot b_r - a_i \cdot b_i) + i(a_r \cdot b_i + a_i \cdot b_r)$$
So it seems we would need 4 multiplications. In general, with a naïve algorithm, it seems we need $(2^n)^2$ multiplications.
MOTIVATION
I believe I've found a faster way to do the multiplications, but maybe there's already a better way. So I'm wondering, how quickly can we multiply two hypercomplex numbers together, modulo a prime?