The Acoustics Black Hole (ABH) effect arouses increasing interests because of the huge potential in damping vibration and reducing sound radiation. Conventional structures with single ABH element usually suffer from the limitation of effective frequency since systematic broadband ABH effect can only be achieved above the characteristics frequency. Previous researches have demonstrated that periodic beams or plates with one directional ABH profiles can achieve directional flexural band gaps at quite low frequencies and thus efficiently low the effective frequency for vibration control. However, complete flexural band gaps have not been revealed among periodic 2D ABH plates due to the complex wave propagation among ABH plates, and thus hamper the low frequency application of ABH plates to some extent. In this paper, we propose a method to construct a kind of periodic plates with 2D ABH indentations. The proposed 2D ABH plates can achieve complete flexural band gaps at low frequency ranges with the combination effect of local resonances and Bragg scattering. By using only a few cells, considerable energy attenuation can be achieved within the corresponding band gaps. The proposed plates show promise for various vibration control applications, especially at low frequencies.