Modular complex field is abelian?

Question

Lets define the set "modular Complex Set modulo prime" as the set of complex numbers {a+bi} (mod p) as {a (mod p) + b(mod p) i}. Is that an abelian group? I'm doing an on-line math course, and this is part of a discussion came up there in the comment-forum. I though that if a certain group is abelian group, such as the set of complex numbers, and we create a new group, by 1:1 mapping from the original one, then the resulting group is also abelian. The numbers a,b in {a+bi} are integers. So, I was not sure that my opinion is right, therefore asked this. The operation on the hypothetical group is comlex number multiplication. For instance: (3+4i)(4+2i) (mod 7) = (4+22i) mod 7 = 4+i, so the group low is complex multiplication.