We investigated the structural stability and magnetic properties of the melt-quenched Heusler alloy-Cu2Mn0.5Fe0.5Al using X-ray diffraction (XRD), scanning electron microscopy (SEM), dc magnetization and ac susceptibility measurements respectively. The structural analysis by Rietveld refinement showed that the alloy crystallised in L21 structure with no trace of secondary phase in the XRD pattern. The dc magnetic measurements exhibited a thermal irreversibility between field-cooled (FC) and zero field cooled (ZFC) magnetization curves at an irreversible temperature Tirr. A deviation from de Almeida –Thouless like line for Tirr vs. H data was observed, suggesting a possibility of cluster glass (CG) like phase in the sample. To confirm the CG behaviour, frequency dependent ac susceptibility (χac) measurements were carried out on the sample and was analysed by Vogel-Fulcher (VF) and Critical slowing down models. We obtained large relaxation time constants (τ0) from the fit to both the phenomenological models which indicate that the origin of glassy nature may not be from randomly oriented atomic spins like in spin-glasses (SG) but of magnetic clusters. The analysis of nonlinear susceptibilities of the sample confirmed the coexistence of ferromagnetic interactions and glassy nature in the sample. Further confirmation for the glassy nature in the sample was provided by ZFC memory experiments. No anomaly was found corresponding to long-range order in the heat capacity (CP(T)) measurement and a T3/2 dependence was also seen in the electrical resistivity which indicates the magnetic glassiness in the sample. © 2018 Elsevier B.V.