multiplication of quaternions is like complex numbers multiplication? Suppose $p = z + j w $ where $z = x_0 + i x_1$ and $w = x_2+ix^3$. Let $q = \alpha + j \beta $ where $\alpha = y_1 + i y_2$ and $\beta = y_2 + i y_3$. How can we multiply $p$ and $q$. Is is just like complex numbers?
$$ pq = (z\alpha - w \beta) + j(\beta w + z \alpha) $$

Hint:
By definition  we  have:
$$
i^2=j^2=k^2=ijk=-1
$$
from which we find:
$$
ij=-ji=k\qquad jk=-kj=i\qquad ki=-ik=j
$$
and with these rules, and the usual rules for multiplication of the real part of the quaternion, you can calculate any product.
