By comparing complex numbers in complex plane and vectors in real plane, we see that they behave 'same'. For example we can find solutions of $z_1+z_2$, $|z_1-z_2|$ and so on by considering $z_i=(x_i,y_i)$ as a vector in real plane.
But I couldn't find the analogous to the product of two complex numbers as vectors in real plane. The definition $(x_1,y_1)(x_2,y_2) = (x_1x_2 - y_1y_2,y_1x_2 + x_1y_2)$ doesn't represent any type of products in real plane, i.e. in case of dot product $(x_1,y_1)(x_2,y_2)=(x_1x_2,y_1y_2)$ and in case of cross product $(x_1,y_1)(x_2,y_2)=(0,0)$ since there is no component in x-y real plane.
What is the vector product comparison of $(x_1,y_1)(x_2,y_2) = (x_1x_2 - y_1y_2,y_1x_2 + x_1y_2)$ in x-y real plane?