2-D complex Gabor filtering has found numerous applications in the fields of computer vision and image processing. Especially, in some applications, it is often needed to compute 2-D complex Gabor filter bank consisting of the 2-D complex Gabor filtering outputs at multiple orientations and frequencies. Although several approaches for fast 2-D complex Gabor filtering have been proposed, they primarily focus on reducing the runtime of performing the 2-D complex Gabor filtering once at specific orientation and frequency. To obtain the 2-D complex Gabor filter bank output, existing methods are repeatedly applied with respect to multiple orientations and frequencies. In this paper, we propose a novel approach that efficiently computes the 2-D complex Gabor filter bank by reducing the computational redundancy that arises when performing the Gabor filtering at multiple orientations and frequencies. The proposed method first decomposes the Gabor basis kernels to allow a fast convolution with the Gaussian kernel in a separable manner. This enables reducing the runtime of the 2-D complex Gabor filter bank by reusing intermediate results of the 2-D complex Gabor filtering computed at a specific orientation. Furthermore, we extend this idea into 2-D localized sliding discrete Fourier transform (SDFT) using the Gaussian kernel in the DFT computation, which lends a spatial localization ability as in the 2-D complex Gabor filter. Experimental results demonstrate that our method runs faster than state-of-the-arts methods for fast 2-D complex Gabor filtering, while maintaining similar filtering quality.