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?
$$ pq = (z\alpha - w \beta) + j(\beta w + z \alpha) $$

Not quite. You also need to remember that for all complex numbers $z=x_0+ix_1$ we have $zj=j\overline{z}$. Therefore the product rule becomes
$$
(z+jw)(\alpha+j\beta)=(z\alpha-\overline{w}\beta)+j(w\alpha+\overline{z}\beta).
$$

Every time you move a complex number to the other side of $j$ you need to replace it with its complex conjugate. It is immaterial whether you move it from left-to-right or right-to-left.
