  We study the correspondence between models of a self-interacting canonical
complex scalar field and P(X)-theories/shift-symmetric k-essence. Both describe
the same background cosmological dynamics, provided that the amplitude of the
complex scalar is frozen modulo the Hubble drag. We compare perturbations in
these two theories on top of a fixed cosmological background. The dispersion
relation for the complex scalar has two branches. In the small momentum limit,
one of these branches coincides with the dispersion relation of the
P(X)-theory. Hence, the low momentum phase velocity agrees with the sound speed
in the corresponding P(X)-theory. The behavior of high frequency modes
associated with the second branch of the dispersion relation depends on the
value of the sound speed. In the subluminal case, the second branch has a mass
gap. On the contrary, in the superluminal case, this branch is vulnerable to a
tachyonic instability. We also discuss the special case of the P(X)-theories
with an imaginary sound speed leading to the catastrophic gradient instability.
The complex field models provide with a cutoff on the momenta involved in the
instability.
