The control plane can be implemented as separate physical hardware or logically as separate processes on a device. There is usually a single control plane (physical or logical) and it hosts both L2 and L3 processes. There is also the concept of distributed control planes where the supervisor handles some tasks and the individual line cards handle other or same tasks. Usually we speak of both L2 and L3 processes belonging to a single control plane on a single device.
When switch stacking technology is used (VSS, FlexStack, IRF), only a single control plane is active for the entire stack. Subordinate switches are on standby and will take over from the master if it fails. The data plane is active on all devices, but the control plane is only fully active on the master device and there is only a single active process for each L2 and L3 feature across the entire stack.
Nexus switches operate differently to normal switch stacking. With Nexus the control planes on both devices are both fully active. Both devices have active control planes for both L2 and L3. When features such as vPC are active, the L2 control planes (one on each device) cooperate to give the impression of a single switch/control plane. In reality there are two separate control planes active and two separate processes for each L2 process. At L3 the control planes act totally independently for most tasks.