I know that a complex number can be taken as 2-dimensional vector and the dot and cross products have also been defined for two complex numbers but different from those of vectors.
But my questions is "What is the vectorial analogue to the usual multiplication of complex numbers? Also the product of two complex numbers is again a complex number"
I think "when it comes to the usual multiplication of complex numbers then complex numbers can't be treated like vectors anymore".
I mean that can we take two vectors and define their multiplication the same way like (a, b)(c, d)= (ac-bd, ad+bc) where ai+bj and ci+dj are two vectors? Please correct me if I am wrong providing a sound exposition.