Results of inelastic neutron scattering measurements are reported for two annealed compositions of Ca(Fe1-xCox)(2)As-2, x = 0.026 and 0.030, which possess stripe-type antiferromagnetically ordered and superconducting ground states, respectively. In the AFM ground state, well-defined and gapped spin waves are observed for x = 0.026, similar to the parent CaFe2As2 compound. We conclude that the well-defined spin waves are likely to be present for all x corresponding to the AFM state. This behavior is in contrast to the smooth evolution to overdamped spin dynamics observed in Ba(Fe1-xCox)(2)As-2, wherein the crossover corresponds to microscopically coexisting AFM order and SC at low temperature. The smooth evolution is likely absent in Ca(Fe1-xCox)(2)As-2 due to the mutual exclusion of AFM ordered and SC states. Overdamped spin dynamics characterize paramagnetism of the x = 0.030 sample and high-temperature x = 0.026 sample. A sizable loss of magnetic intensity is observed over a wide energy range upon cooling the x = 0.030 sample, at temperatures just above and within the superconducting phase. This phenomenon is unique amongst the iron-based superconductors and is consistent with a temperature-dependent reduction in the fluctuating moment. One possible scenario ascribes this loss of moment to a sensitivity to the c-axis lattice parameter in proximity to the nonmagnetic collapsed tetragonal phase and another scenario ascribes the loss to a formation of a pseudogap.