I just have to prove that it isn't with O(A+B)=O(A)+O(B) and O(kA)=k(OA) where O is the linear operator (i.e the absolute value), A+B and A would be a complex number, and k is some real constant. I used abs(x+iy). sooo when you break it up it would be abs(x)+abs(iy) which isn't true...right? Or am I completely thinking of this the wrong way?
Yes, you are thinking in a right way. (there are lots of ways to think about this)
To make it more precise, if the absolute value were a linear operator, then
$$ |x + iy| = |x| + |iy|$$
would be an identity. (aside: if this were an identity, that's still not enough to prove the absolute value a linear operator)
So to prove that the absolute value is not a linear operator, it is enough to prove that the equation above is not an identity.
Note it's not enough to just look at them and say they look different: sometimes you can have very different expressions that nonetheless represent the same thing. One simple way to prove that this isn't an identity is to plug in some specific values and see that they are unequal.