1. Let $z = e^{i\theta}$ since $|z|=1$.

   Then $\bar z = e^{-i\theta}.$

2. Compute the given expression: $\frac{z}{\bar z} = \frac{e^{i\theta}}{e^{-i\theta}} = e^{2i\theta}, \qquad \frac{\bar z}{z} = e^{-2i\theta}.$

   So, $\frac{z}{\bar z}+\frac{\bar z}{z} = e^{2i\theta}+e^{-2i\theta} = 2\cos 2\theta.$

3. Apply the condition $\left|\frac{z}{\bar z}+\frac{\bar z}{z}\right|=1.$

   Hence, $|2\cos 2\theta|=1$ $|\cos 2\theta|=\frac12.$

4. Solve $\cos 2\theta = \pm \frac12.$

   In $[0,2\pi)$, the values of $2\theta$ satisfying this are: $2\theta = \frac{\pi}{3},\ \frac{2\pi}{3},\ \frac{4\pi}{3},\ \frac{5\pi}{3},\ \frac{\pi}{3}+2\pi,\ \frac{2\pi}{3}+2\pi,\ \frac{4\pi}{3}+2\pi,\ \frac{5\pi}{3}+2\pi$ because $2\theta$ ranges over $[0,4\pi)$ when $\theta\in[0,2\pi)$.

   Dividing by $2$, we get 8 distinct values of $\theta$: $\theta = \frac{\pi}{6},\ \frac{\pi}{3},\ \frac{2\pi}{3},\ \frac{5\pi}{6},\ \frac{7\pi}{6},\ \frac{4\pi}{3},\ \frac{5\pi}{3},\ \frac{11\pi}{6}.$

5. Therefore, the number of complex numbers $z$ is $8.$

6. Checking options:

   • A: $8$ ✅
   • B: $10$ ❌
   • C: $4$ ❌
   • D: $6$ ❌

So the correct option is A.