Both catalogs are proprietary. But depending on the data (for example DNGs that contain a lot of non proprietary data that is also stored in a catalog), a lot can be imported either way. The LR catalog stores the location of the images and folder structure so if you have a well organized structure, LR will fully support that. All proprietary raw process or parametric edits will not transfer either way; they are proprietary.
So: No you can't import the C1 catalog itself. But no, you don't have to start totally from scratch.