We used resonant inelastic x-ray scattering to reveal the nature of magnetic interactions in Sr2IrO4, a 5d transition-metal oxide with a spin-orbit entangled ground state and J(eff)=1/2 magnetic moments. The magnon dispersion in Sr2IrO4 is well-described by an antiferromagnetic Heisenberg model with an effective spin one-half on a square lattice, which renders the low-energy effective physics of Sr2IrO4 much akin to that in superconducting cuprates. This point is further supported by the observation of exciton modes in Sr2IrO4, whose dispersion is strongly renormalized by magnons, which can be understood by analogy to hole propagation in the background of antiferromagnetically ordered spins in the cuprates.