Let $$S= \left\{z \in \mathbb{C}: 4z^{2}+ \overline{z}=0 \right\}$$. Then $$\sum_{z\in S} |z|^{2}$$ is equal to:

 $$4z^2 = -\bar{z}$$.

$$|4z^2| = |-\bar{z}| \implies 4|z|^2 = |z|$$

This gives two possibilities for $$|z|$$:

o $$|z| = 0 \implies z = 0$$ (This is one solution).

o $$4|z| = 1 \implies |z| = \frac{1}{4}$$ (For non-zero solutions).

If $$|z| = \frac{1}{4}$$, then $$|z|^2 = \frac{1}{16}$$.

From the original equation $$4z^2 = -\bar{z}$$, multiply both sides by $$z$$:

$$4z^3 = -z\bar{z} = -|z|^2 = -\frac{1}{16}$$

$$z^3 = -\frac{1}{64}$$

This is a cubic equation, which provides 3 distinct roots, all having the same magnitude $$|z| = \frac{1}{4}$$.

The set $$S$$ contains the root $$z=0$$ and the 3 roots where $$|z|^2 = \frac{1}{16}$$.

$$\sum_{z \in S} |z|^2 = 0^2 + \left(\frac{1}{16} + \frac{1}{16} + \frac{1}{16}\right) = \mathbf{\frac{3}{16}}$$

Correct Option: B