Embedding colloidal particles in hydrogels endows the polymer with appealing characteristics for microfluidics and biosensing applications. This theoretical study was motivated by the possibility of developing new diagnostic methods for probing the dynamics of hydrogel composites using electric fields. While electrical microrheolgy presents several difficulties, the technique could sense physicochemical characteristics of the particle-polymer interface that are beyond the reach of existing magnetic and passive microrheology techniques. In this work, we present the first calculations of electric-field-induced colloid displacement in compressible polymer gels under steady and harmonic electrical forcing. In contrast to Hill and Ostoja-Starzewski's theory for the steady displacement of particles in incompressible hydrogels (Phys. Rev. E, 2008), we show that polymer compressibility leads to a particle displacement that grows linearly with particle size when the Debye length is smaller than the particle radius; moreover, the particle displacement is significantly larger than in incompressible gels (Wang & Hill, Soft Matter, 2008), which suggests the theory could be tested using optical microscopy. Under oscillatory forcing, which is particularly relevant to electroacoustic diagnostics, the amplitude of the fluctuating colloid displacement exhibits a transition from quasi-steady compressible to quasi-steady incompressible dynamics as the frequency passes through the reciprocal draining time of the gel. Our calculations demonstrate how the frequency response depends on the particle size and charge, ionic strength of the electrolyte, and elastic and hydrodynamic characteristics of the polymer network.