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The 946nm diode-pump microchip self-Q-switched laser of a chromium and neodymium codoped yttrium aluminum gaet crystal material (Cr4+Nd3+:YAG) is studied, especially about its physical mechanism of operation. The 4F3/2→ 4I9/2 transition of Nd3+ ion is beneficial to achieving laser oscillation in a quasi-three-level system based on coating the cavity mirrors of the microchip with films that suppress the 1064nm operation and enhance the 946nm laser. The Cr4+ ion is a saturable absorber. The initial loss Nt1 is high, which acts as the threshold for laser oscillation. The stable loss Nt2 is low because the Cr4+ ion is acceleratively bleached by the fast enhancement of the oscillating laser. The high Nt1, small Nt2 and fast progresses permit the oscillating laser of the Cr4+Nd3+:YAG to have a good self-Q-switched property whose full width at half maximum is about 4.2ns. Its highest laser power is about 5.7mW. Its peak power is about 150W. Its good fundamental transverse TEM00 mode results from the absorption bleaching established by both the pump and oscillating lasers, which suppress other transverse mode and allow the oscillation only in the fundamental transverse TEM00 mode.