We have studied the short-time dynamical behavior of polydimethylsiloxane (PDMS) bottlebrush polymers, PDMS-g-PDMS. The samples have similar backbone lengths but different side-chain lengths, resulting in a shape transition. Quasi-elastic neutron scattering was used to observe the dynamical changes inherent to these structural changes. The combination of data from three spectrometers enabled to follow the dynamics over broad frequency and temperature ranges, which included segmental relaxations and more localized motions. The latter, identified as the methyl group rotation, is described by a threefold jump model and shows higher activation energies compared to linear PDMS. The segmental relaxation times, tau(s) decrease with increasing molecular weight of the side chains but increase with momentum transfer, Q, following a power law, which suggests a non-Gaussian behavior for bottlebrush polymers.