Is there any way to "follow" a user on Slack? That is, be notified whenever this user posts something publicly?
I did not find anything here, in the Slack docs, and https://productivityist.com/7-things-about-slack/ .
Slack does not have a build-in "follow" feature. If you need this feature I would advise to send a mail to the Slack team. They are very helpful and are open to add new features based on user input. One straightforward way to implement this for the Slack team would be to add users to notification settings, similar to keywords.
Or you can implement it with your own Slack app. One approach would be to use the Events API for monitoring all messages and then add ephemeral pings to a channel whenever a followed user posts a message in that channel.