Add category image with REST

At the moment this is not possible without extending core functionality. However, category image save/get support can be added to category REST API using extension attributes mechanism:

1. Declare extension attribute of string type for \Magento\Catalog\Api\Data\CategoryInterface (it will hold base64-encoded image)
2. Write afterSave plugin for \Magento\Catalog\Api\CategoryRepositoryInterface::save to save image
3. Write afterLoad plugin for \Magento\Catalog\Api\CategoryRepositoryInterface::get to load image

See more detailed instructions on declaring extension attributes in this answer