How compatible?
Can I just expect things like full text search to work? https://www.postgresql.org/docs/13/textsearch.html
What about additional but supplied modules like ltree? https://www.postgresql.org/docs/current/ltree.html
I ask as I saw a related article https://www.cockroachlabs.com/blog/full-text-indexing-search... recently and damn it looks close... has anyone migrated a prod system that uses the above? What did you encounter?
Be very careful - even if things are compatible, in my experience some things do not perform as well in CockroachDB which seems a bit counter intuitive but can be very true... We had problems making date range queries fast with tiny amounts of data, for example.
I would seriously consider if you need CockroachDB - if you need that level of scaling you should also consider Cassandra (or Cassandra like) solutions as they will scale better but you will have to architect your app to think in this way (i.e. your app generates the views of the data it needs).
If you don't need to scale yet use Postgres. Every piece of complexity has a cost and Cockroach while incredibly clever adds complexity that you might not understand or desire to manage over time. A nice interface won't help you when replication between clusters breaks in production.