Motivated by the chain antiferromagnet BaCo2V2O8, we study the excitation spectrum of the one-dimensional spin-1/2 XXZ chain across a magnetic quantum phase transition induced by the application of a site-dependent transverse magnetic field. Considering a situation where the transverse magnetic field has a strong uniform component and a weaker staggered part, we determine the nature of the excitations giving rise to the spin dynamical structure factor using a combination of analytical approaches and the numerically exact time-dependent matrix product state method. Below the quantum phase transition, we identify high-energy many-body two-magnon and three-magnon repulsively bound states which are clearly visible due to the staggered component of the magnetic field. At high magnetic fields and low temperature, single magnons dominate the dynamics. These results are in very good agreement with terahertz spectroscopy measurements.