I'm not quite sure what you're asking, but I'll try to answer you anyway.

If a is a complex number, then a can be written as a=x+iy where x and y are real numbers. The conjugate of a is a*=x-iy. A complex number a is real if and only if a=a* (or equivalently, y=0).

If you want to make sense of what b is in terms of real numbers, then you need to let c=x+iy and d=z+iw for real numbers x,y,z,w and then simplify those expressions.

Saying y=(cd*-c*d) doesn't make sense until you are sure that cd*-c*d is real (which I believe it is, but it must be checked first).

I'm not sure if I understood your question but I hope some of this helps.