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We report on a compact, highly sensitive all-fiber accelerometer suitable for low frequency and low amplitude vibration sensing. The sensing elements in the device are two short segments of strongly coupled asymmetric multicore fiber (MCF) fusion spliced at 180{\deg} with respect to each other. Such segments of MCF are sandwiched between standard single mode fibers. The reflection spectrum of the device exhibits a narrow spectrum whose height and position in wavelength changes when it is subjected to vibrations. The interrogation of the accelerometer was carried out by a spectrometer and a photodetector to measure simultaneously wavelength shift and light power variations. The device was subjected to a wide range of vibration frequencies, from 1 mHz to 30 Hz, and accelerations from 0.76 mg to 29.64 mg, and performed linearly, with a sensitivity of 2.213 nW/mg. Therefore, we believe the accelerometer reported here may represent an alternative to existing electronic and optical accelerometers, especially for low frequency and amplitude vibrations, thanks to its compactness, simplicity, cost-effectiveness, implementation easiness and high sensitivity.