I can't answer all of your questions, but I can tell you how I would attempt to do it.
a + bi is made from 3 components, real numbers (a, b), operators (+) and the imaginary symbol (i). So we need to be able to express all 3. + and i are easy, it's just the symbol themselves.
So onto a real number, we can either write its decimal expansion (4.2) or with algebraic symbols (sqrt(3)/2). One of these is much simpler, but since you included sqrt, is say you need the latter.
Let's build up and see where we get to. 2 is easy it's just [0-9]. 42? [0-9]+
sqrt(-7)? Oh, we don't want to be able to show, since that wouldn't be a real number.
But they are the same? Oh, we made a mistake, we forgot about parenthesis. From the pumping lemma, we can't describe parenthesis that are matched, so this definition is not a regular language. :/
Fine, lets pick the first way of expressing real numbers. I'm going to gloss over the steps here, since it is a quick google away.
(-)[0-9]+(.[0-9]+) where we escaped the . So that we know it's just a dot and not any single symbol.
Thus we say we can represent a complex number by
Notice the second choice is so we can write 21, or -5i, but we can also write 3i and not be forced into +3i