How to do complex number multiplication using only three real number multiplication

You are interested in two numbers : α1=ac−bd and α2=ad+bc.
You can compute 3 products, viz. P1=ac,P2=bd, and P3=(a+b)(c+d). Then α1=P1−P2, and α2=P3−P2−P1.
This process is often called Karatsuba multiplication, and is used in algorithm design quite frequently.