Let $\omega$ be a complex number such that $|\omega| = 1,$ and the equation
\[z^2 + z + \omega = 0\]has a pure imaginary root $z.$  Find $\omega + \overline{\omega}.$
Let the pure imaginary root be $ki,$ where $k$ is real, so
\[-k^2 + ki + \omega = 0.\]Thus, $\omega = k^2 - ki.$  Then $\overline{\omega} = k^2 + ki,$ so
\[1 = |\omega|^2 = \omega \overline{\omega} = (k^2 - ki)(k^2 + ki) = k^4 + k^2.\]Then $k^4 + k^2 - 1 = 0.$  By the quadratic formula,
\[k^2 = \frac{-1 \pm \sqrt{5}}{2}.\]Since $k$ is real,
\[k^2 = \frac{-1 + \sqrt{5}}{2}.\]Therefore,
\[\omega + \overline{\omega} = k^2 - ki + k^2 + ki = 2k^2 = \boxed{\sqrt{5} - 1}.\]