Prahlad Vaidyanathan has already answered on how performing the product between two complex numbers with only three real multiplications. Now the question is: can we perform the product between two complex numbers with less than three real multiplications?
The answer is NO and is provided by Winograd's theorem in
S. Winograd, "On the number of multiplications required to compute certain functions", Commun. Pure Appl. Math. 23 (1970), 165-179.
The minimum number of multiplications required in the computation of the product between two complex numbers is three.