Let $$S = \{z \in \mathbb{C} : \bar{z}^2 + \bar{z} = 0\}$$. Then $$\sum_{z \in S} (\text{Re}(z) + \text{Im}(z))$$ is equal to______.

Given,

$$S=\{z\in\mathbb C:\bar z^2+\bar z=0\}$$

We have

$$\bar z^2+\bar z=0$$

$$\bar z(\bar z+1)=0$$

Hence,

$$\bar z=0\quad \text{or}\quad \bar z=-1$$

Taking conjugate on both sides,

$$z=0\quad \text{or}\quad z=-1$$

Thus,

$$S=\{0,-1\}$$

Now compute

$$\sum_{z\in S}(\operatorname{Re}(z)+\operatorname{Im}(z))$$

For

$$z=0,$$

$$\operatorname{Re}(0)+\operatorname{Im}(0)=0$$

For

$$z=-1,$$

$$\operatorname{Re}(-1)+\operatorname{Im}(-1)=-1+0=-1$$

Therefore,

$$0+(-1)=-1$$

Hence, the required value is

$$\boxed{-1}$$.