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,
\(\omega + \overline{\omega} = k^2 - ki + k^2 + ki = 2k^2 = \boxed{\sqrt{5} - 1}.\)
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