  We have studied the magnetic excitations of electron-doped
Sr$_{2-x}$La$_x$IrO$_4$ ($0 \leq x \leq 0.10$) using resonant inelastic x-ray
scattering (RIXS) at the Ir L$_3$-edge. The long range magnetic order is
rapidly lost with increasing $x$, but two-dimensional short-range order (SRO)
and dispersive magnon excitations with nearly undiminished spectral weight
persist well into the metallic part of the phase diagram. The magnons in the
SRO phase are heavily damped and exhibit anisotropic softening. Their
dispersions are well described by a pseudospin-1/2 Heisenberg model with
exchange interactions whose spatial range increases with doping. We also find a
doping-independent high-energy magnetic continuum, which is not described by
this model. The spin-orbit excitons arising from the pseudospin-3/2 manifold of
the Ir ions broaden substantially in the SRO phase, but remain largely
separated from the low-energy magnons. Pseudospin-1/2 models are therefore a
good starting point for the theoretical description of the low-energy magnetic
dynamics of doped iridates.
