  We initiate the study of stability and pseudospectra of complex momentum
modes of asymptotically anti-de Sitter black holes. Similar to quasinormal
modes, these can be defined as the poles of the holographic Green's function,
albeit for real frequency and complex momentum. Their pseudospectra are in
stark contrast to the pseudospectra of quasinormal modes of AdS black holes.
Contrary to the case of quasinormal mode pseudospectra, the resolvent is
well-defined, and the numerical approximation shows fast convergence. At zero
frequency, complex momentum modes are stable normal modes of a Hermitian
operator. Even for large frequencies, they show only comparatively mild
spectral instability. We also find that local potential perturbations cannot
destabilize the lowest complex momentum mode.
