In condensed matter systems, out of a large number of interacting degrees of freedom emerge weakly coupled quasiparticles (QPs), in terms of which most physical properties are described. The lack of identification of such QPs is a major barrier for understanding myriad exotic properties of correlated electrons, such as unconventional superconductivity and non-Fermi liquid behaviours. Here we report the observation of a composite particle in a quasi-two-dimensional spin-1/2 antiferromagnet Sr₂ IrO₄ - an exciton dressed with magnons - that propagates with the canonical characteristics of a QP: a finite QP residue and a lifetime longer than the hopping time scale. The dynamics of this charge-neutral excitation mirrors the fundamental process of the analogous one-hole propagation in the background of spins-1/2, and reveals the same intrinsic dynamics that is obscured for a single, charged-hole doped into two-dimensional cuprates.