© 2015 American Physical Society. We present the results of elastic and inelastic neutron scattering measurements on nonsuperconducting Ba(Fe0.957Cu0.043)2As2, a composition close to a quantum critical point between antiferromagnetic (AFM) ordered and paramagnetic phases. By comparing these results with the spin fluctuations in the low-Cu composition as well as the parent compound BaFe2As2 and superconducting Ba(Fe1-xNix)2As2 compounds, we demonstrate that paramagnon-like spin fluctuations are evident in the antiferromagnetically ordered state of Ba(Fe0.957Cu0.043)2As2, which is distinct from the AFM-like spin fluctuations in the superconducting compounds. Our observations suggest that Cu substitution decouples the interaction between quasiparticles and the spin fluctuations. We also show that the spin-spin correlation length ξ(T) increases rapidly as the temperature is lowered and find ω/T scaling behavior, the hallmark of quantum criticality, at an antiferromagnetic quantum critical point.