Why can't be a number the gcd of two numbers?

Question

I have the ring $R = \mathbb{Z[i\sqrt{3}]}$, and I need to find the gcd of $a = 6+2i\sqrt{3}, b = 4-2i\sqrt{3}$. This is the way I solved it:

Let $d = x+yi\sqrt{3}$ be $gcd(a,b)$ then $d |a$ and $d|b$ and also $N(d)|N(a) = 48$ and $N(d)|N(b) = 28$ so $N(d)| 4$. Therefore $d$ could be + or -$1,2$ or $1+i\sqrt3$ because these numbers have norm that divide 4. So I would say that gcd is $1+i\sqrt3$. But it turns out that this can't be gcd because it is not dividing a and b.

So my question is why can't $1+i\sqrt3$ be gcd?