Checking up the definition of (a+ib)(c+id), it's
(ac-bd) + i (ad+bc)

which is what bwkaz found.

However, the question was:
Compute the complex product ( a + ib ) * ( c + id ) using only three real multiplications

We've only used 1 multiplication, and that's multiplying (ad+bc) by i.
The only further step I could find was:

i((1/i)(ac-bd) + (ad+bc))

but nothing else. Any other takers?

After I was so pleased with my web server