  This study examines the relationship between magnetic field complexity and
mass and angular momentum losses. Observations of open clusters have revealed a
bimodal distribution of the rotation periods of solar-like stars that has
proven difficult to explain under the existing rubric of magnetic braking.
Recent studies suggest that magnetic complexity can play an important role in
controlling stellar spin-down rates. However, magnetic morphology is still
neglected in most rotation evolution models due to the difficulty of properly
accounting for its effects on wind driving and angular momentum loss. Using
state-of-the-art magnetohydrodynamical magnetized wind simulations we study the
effect that different distributions of the magnetic flux at different levels of
geometrical complexity have on mass and angular momentum loss rates. Angular
momentum loss rates depend strongly on the level of complexity of the field but
are independent of the way this complexity is distributed. We deduce the
analytical terms representing the magnetic field morphology dependence of mass
and angular momentum loss rates. We also define a parameter that best
represents complexity for real stars. As a test, we use these analytical
methods to estimate mass and angular momentum loss rates for $8$ stars with
observed magnetograms and compare them to the simulated results. Magnetic field
complexity provides a natural physical basis for stellar rotation evolution
models requiring a rapid transition between weak and strong spin-down modes.
