Using the decomposition of an image field in two spatial components that can be controllably shifted in phase with respect to each other, a new quantitative-phase microscope has been developed. The new instrument, referred to as the fast Fourier phase microscope (f-FPM), provides a factor of 100 higher acquisition rate compared with our previously reported Fourier phase microscope. The resulting quantitative-phase images are characterized by diffraction limited transverse resolution and path-length stability better than 2\text{\hspace{0.17em} nm} at acquisition rates of 10 frames∕s or more. These features make the f-FPM particularly appealing for investigating the structure and dynamics of live cells over a broad range of time scales. In addition, we demonstrate the possibility of examining subcellular structures by digitally processing the amplitude and phase information provided by the instrument. Thus we developed software that can emulate phase contrast and differential interference contrast microscopy images by numerically processing FPM images. This approach adds the flexibility of digitally varying the phase shift between the two interfering beams. The images obtained appear as if they were recorded by variable phase contrast or differential interference contrast microscopes that deliver an enhanced view to the subcellular structure when compared with the typical commercial microscope.