Let $S=\{z \in \mathbb{C} \mid z^5=1\}$
We know that, $$S=\left\{\cos(\frac{2\pi j}{5})+\iota \sin(\frac{2\pi j}{5})\mid j=0,1,2,3,4\right\}$$ and that $S$ is a group w.r.t. multiplication of complex numbers. So $1 \in S$.
Under Addition
Since $1 \in S$, $1+1=2$ must be in $S$. But we know that $2^5=32 \neq 1$. This implies that $2 \notin S$ and hence $S$ is not closed under addition and consequently not a group under addtion.
Please let me know if my conclusion and\or approach is right.
That's fine. One counterexample, even using the same value twice, is enough. Indeed, since you've refuted closure, this set doesn't even form a magma under addition.