Sure. So far, we just precompiled a few popular extensions, and they are available for installation. Ultimately we want to provide an option to bring your own extensions e.g., by specifying a docker image based on our base image. But that is some work on a security front: with a custom extension, you have access to the corresponding Unix user and can construct malicious WAL, send requests to the control plane, etc.

Disclaimer: Neon co-founder

I'm guessing no, or at most just have set of extensions that you can use (kind of like it's done in AWS RDS or AWS Aurora).

The claim is "serverless" i.e. you don't have access to the server and if you could install any extension you essentially have full access to the server as there's no restriction what you could do in an extension. I don't think that would be allowed.

https://neon.tech/docs/cloud/compatibility/ says "During technical preview Neon has restrictions on user ability to install PostgreSQL extensions. Following PostgreSQL extensions come pre-installed: [..]"