  In this paper by making use of the "Complexity=Action" proposal, we study the
complexity growth after shock waves in holographic field theories. We consider
both double black hole-Vaidya and AdS-Vaidya with multiple shocks geometries.
We find that the Lloyd's bound is respected during the thermalization process
in each of these geometries and at the late time, the complexity growth
saturates to the value which is proportional to the energy of the final state.
We conclude that the saturation value of complexity growth rate is independent
of the initial temperature and in the case of thermal initial state, the rate
of complexity is always less than the value for the vacuum initial state such
that considering multiple shocks it gets more smaller. Our results indicate
that by increasing the temperature of the initial state, the corresponding rate
of complexity growth starts far from final saturation rate value.
