I know that $K=\Bbb Q_2(\sqrt{-3})$ is the only unramified quadratic extension of the 2- adic field $\Bbb Q_2$.
How many roots of unity exists in $K=\Bbb Q_2(\sqrt{-3})$ and in what order?
Clearly $\pm 1$ is in $K=\Bbb Q_2(\sqrt{-3})$. So $2$ roots of unity is here. Is there any more?
Next, what is the order of the roots of power of $2$?
I'll put here what I left as a comment above.
If $k$ is the residue field of $K$ of characteristic $p$, then there is an isomorphism (of groups) $k^{\times} \rightarrow \mu_K^{(p)}$, where $\mu_K^{(p)}$ is the set of roots of unity in $K$ whose order is not divisible by $p$. (This isomorphism is given by the Teichmuller map).
So all that remains is to check the roots of unity of two power order (in fact it suffices to show $i=\sqrt{-1} \not\in K$ which you can do via ramification theory).