The stellar winds of the nearby binary system alpha Cen A and B interact strongly with each other, leading to a structure (a collision disk) outlined by strong shocks and regions of high density and temperature. Further away, the combined winds encounter a moderately dense, warm, and partially ionized interstellar medium with which they interact as well, forming an astrosphere. By the nature of the binary orbital motion, the resulting astrosphere is strongly time dependent on multiple scales. A comprehensive numerical modeling study in 3D that treats plasma and neutrals separately while allowing for the effects of charge exchange between these species, is presented. Observables like X-rays from the stellar wind charge exchange emission mechanism are discussed, and the thermodynamically distinct regions inside the alpha Cen AB astrosphere are characterized. These time dependent regions will have to be taken into account in the assessment of the evolution and habitability of any exoplanet that this system may harbor.