
The cross procuct actually is something in three dimensions. The cross product of (x1, y1, z1) and (x2, y2, z2) is (y1 * z2  y2 * z1, z1 * y2  z2 * y1, x1 * y2  x2 * y1). Now if we see our two vectors as three dimensional vectors with z = 0, the cross product will have x = 0 and y = 0 and z = x1 * y2  x2 * y1.
But looking at the part you quoted, shouldn't that be zw = (a + bi) * (c + di) = ac + adi + bci + bdii = (ac  bd) + (ad + bc)i? So that the real part is the cross product and the imaginary part is the dot product, instead of the other way around? 