The time evolution of strongly excited SU(2) Bogomol'nyi-Prasad-Sommerfield magnetic monopoles in Minkowski spacetime is investigated by using numerical simulations based on the technique of conformal compactification and on the use of the hyperboloidal initial value problem. It is found that an initially static monopole does not radiate the entire energy of the exciting pulse toward future null infinity. Rather, a long-lasting quasistable "breathing state" develops in the central region and certain expanding shell structures-built up by very high frequency oscillations-are formed in the far away region.