From the datasheet, it looks like the C3 doesn't have the Ultra Low Power (ULP) co-processor. This co-processor was there on previous ESP32s and allowed you to run code for sensor monitoring or other basic tasks while the main CPU was stopped, consuming ~150µA. It can then wake the main CPU if certain conditions were met. Wonder why it was removed, and if this would prevent the C3 from being used in these kinds of low-power applications.

Datasheet: https://www.espressif.com/sites/default/files/documentation/...