The surprise is that the definition you gave in 2. fully agrees with the definition of multiplication that comes naturally from the cartesian version of complex numbers: z = x + bi -> (x1 + b1i)(x2 + b2i) = x1x2 + x1b2i + x2b1i - b1b2

Vector multiplication has two very different forms (dot and cross), so multiplication (and by extension exponentiation) working out smoothly is neat.