Compressive spectral depth imaging (CSDI) is an emerging technology aiming to reconstruct spectral and depth information of a scene from a limited set of two-dimensional projections. CSDI architectures have conventionally relied on stereo setups that require the acquisition of multiple shots attained via dynamically programmable spatial light modulators (SLM). This work proposes a snapshot CSDI architecture that exploits both phase and amplitude modulation and uses a single image sensor. Specifically, we modulate the spectral-depth information in two steps. Firstly, a deformable mirror (DM) is used as a phase modulator to induce a focal length sweeping while simultaneously introducing a controlled aberration. The phase-modulated wavefront is then spatially modulated and spectrally dispersed by a digital micromirror device (DMD) and a prism, respectively. Therefore, each depth plane is modulated by a variable phase and binary code. Complimentary, we also propose a computational methodology to recover the underlying spectral depth hypercube efficiently. Through simulations and our experimental proof-of-concept implementation, we demonstrate that the proposed computational imaging system is a viable approach to capture spectral-depth hypercubes from a single image.