We report ALMA Early Science CO(1-0) and CO(3-2) observations of the brightest cluster galaxy (BCG) in A1664. The BCG contains 1.1 × 1010 M ☉ of molecular gas divided roughly equally between two distinct velocity systems: one from –250 to +250 km s–1 centered on the BCG's systemic velocity and a high-velocity system blueshifted by 570 km s–1 with respect to the systemic velocity. The BCG's systemic component shows a smooth velocity gradient across the BCG center, suggestive of rotation about the nucleus. However, the mass and velocity structure are highly asymmetric and there is little star formation coincident with a putative disk. It may be an inflow of gas that will settle into a disk over several 108 yr. The high-velocity system consists of two gas clumps, each ~2 kpc across, located to the north and southeast of the nucleus. Each has a line of sight velocity spread of 250-300 km s–1. The velocity of the gas in the high-velocity system increases toward the BCG center and may be a massive flow into the nucleus. However, the velocity gradient is not smooth. These structures are also coincident with low optical-ultraviolet surface brightness regions, which could indicate dust extinction associated with each clump. The structure is complex, making a clear interpretation difficult, but if the dusty, molecular gas lies predominantly in front of the BCG, the blueshifted velocities would indicate an outflow. Based on the energy requirements, such a massive outflow would most likely be driven by the active galactic nucleus. A merger origin is unlikely but cannot be ruled out.