, 2015

Solution:
Since $|z| = |\bar{z}|$ we may divide by $|z|$ and assume that $|z| = 1$. Then $\bar{z} = \frac{1}{z}$, so we are looking for the smallest positive integer $n$ such that there is a $2n^{\text{th}}$ root of unity in the first quadrant. Clearly there is a sixth root of unity in the first quadrant but no fourth or second roots of unity, so $n = 3$ is the smallest.