Does movement along the imaginary axis of a root locus change the natural frequency?
If a system has a pole at \$s = a + ib\$, then it has a response that looks like
$$
r(t) = e^{(a + ib)t} = e^{at}e^{ibt}
$$
If these poles have a complex part (ie: \$b \neq 0\$), then the pair of poles create a real response that looks like
$$
r(t) = e^{at} \sin(bt) \text{ or } e^{at} \cos(bt)
$$

To answer your question:


Moving the poles along the real axis doesn't change the response frequency. Instead, it changes how quickly the response decays.
Moving the poles along the imaginary axis changes how fast the response oscillates.


So, like the author suggests, if the imaginary part of the roots doesn't change, then the frequency doesn't change either.