hi,
you have to use computer arithmetic ,not mathematic. in digital world, only adder exist. so you have to implement your algorithm to adder&other control logic.
there are so many books describing "computer arithmetic", you can find them as reference and very useful.
usually people transform complex number operation into real one. For example: multiplying (a + jb) * (c + jd) = (ac - bd) + j(bc + ad), so you save the first complex number into two real numbers, and the second complex number also into two real numbers, then the result of the multiplication can be calculated based on those 4 real numbers.