We use the complexity=volume (CV) prescription to study the effect of a magnetic field on the computational complexity for states in the gauge theories dual to two different gravitational models. In one of these theories the complexity increases with the intensity of the magnetic field, while in the other a more interesting behavior is discovered, resulting in a phenomenon that we term magnetic simplification. The relevant difference between the two theories is that the content of the second includes a scalar operator with a nonvanishing vacuum expectation value. This leads us to conclude that the direct impact of the magnetic field is to increase the complexity of a state, but it can indirectly lower it by diminishing the complexity associated to additional degrees of freedom when these do not vanish across the space. We additionally compare the results obtained working in the full ten-dimensional backgrounds and in their effective five-dimensional truncations, exhibiting that the question is still current about which surface, whether the uplift of the 5D extremal hypersurface or the extremal surface in 10D, should be used in the CV prescription.

17 pages, 15 figures. Published version, updated figures 4,6,12 and some extra comments about these were added