  Rapidly spinning, strongly magnetized proto-neutron stars ("millisecond
proto-magnetars") are candidate central engines of long-duration gamma-ray
bursts (GRB), superluminous supernovae (SLSNe), and binary neutron star
mergers. Magnetar birth may be accompanied by the fall-back of stellar debris,
lasting for seconds or longer following the explosion. Accretion alters the
magnetar evolution by (1) providing an additional source of rotational energy
(or a potential sink, if the propeller mechanism operates); (2) enhancing the
spin-down luminosity above the dipole rate by compressing the magnetosphere and
expanding the polar cap region of open magnetic field lines; (3) supplying an
additional accretion-powered neutrino luminosity that sustains the wind
baryon-loading, even after the magnetar's internal neutrino luminosity has
subsided. The more complex evolution of the jet power and magnetization of an
accreting magnetar more readily accounts for the high 56Ni yields GRB SNe and
irregular time evolution of some GRB light curves (e.g.~bursts with precursors
followed by a long quiescent interval before the main emission episode).
Additional baryon-loading from accretion-powered neutrino irradiation of the
magnetar polar cap lengthens the timeframe over which the jet magnetization is
in the requisite range sigma ~< 1e3 for efficient gamma-ray emission, thereby
accommodating GRBs with ultra-long durations. Though accretion does not
significantly raise the maximum energy budget from the limit of <~ few 1e52
ergs for an isolated magnetar, it greatly expands the range of magnetic field
strengths and birth spin periods capable of powering GRB jets, reducing the
differences between the magnetar properties normally invoked to explain GRBs
versus SLSNe.
