Hyperaccreting neutron star or magnetar disks cooled via neutrino emission can be a candidate of gamma-ray burst (GRB) central engines. The strong field ≥ 10 15 − 10 16 G of a magnetar can play a significant role in affecting the disk properties and even lead to the funnel accretion process. In this paper we investigate the effects of strong fields on the disks around magnetars, and discuss implications of such accreting magnetar systems for GRBs and GRB-like events. We discuss quantum effects of the strong fields on the disk thermodynamics and microphysics due to modifications of the electron distribution and energy in the strong field environment, and use the magnetohydrodynamical conservation equations to describe the behavior of the disk flow coupled with a large scale field, which is generated by the star-disk interaction. If the disk field is open, the disk properties mainly depend on the ratio between |B φ /B z | and Ω/Ω K with B φ and B z being the azimuthal and vertical components of the disk field, Ω and Ω K being the accretion flow angular velocity and Keplerian velocity respectively. On the other hand, the disk properties also depend on the magnetar spin period if the disk field is closed. In general, stronger fields give higher disk densities, pressures, temperatures and neutrino luminosity. Moreover, strong fields will change the electron fraction and degeneracy state significantly. A magnetized disk is always viscously stable outside the Alfvén radius, but will be thermally unstable near the Alfvén radius where the magnetic field plays a more important role in transferring the angular momentum and heating the disk than the viscous stress. The funnel accretion process will be only important for an extremely strong field, which creates a magnetosphere inside the Alfvén radius and truncates the plane disk. Because of higher temperature and more concentrated neutrino emission of a ringlike belt region on the magnetar surface covered by funnel accretion, the neutrino annihilation rate from the accreting magnetar can be much higher than that from -2an accreting neutron star without fields. Furthermore, the neutrino annihilation mechanism which releases the gravitational energy of the surrounding disk and the magnetically-driven pulsar wind which extracts the stellar rotational energy from the magnetar surface can work together to generate and feed an ultrarelativistic jet along the stellar magnetic poles.