At the moment any chat user can ignore a chat moderator† and not see their messages anymore.
Should this be the case? After all:
a moderator needs to be able to communicate with all the users of their community. Chat is an important medium and in many cases a fundamental tool for moderation
ignoring a moderator does not mean that the user is not subject to moderation, a user needs to be able to see moderator messages
there's nothing preventing a user from ignoring the whole moderation team
when there are cases of moderator abuse, ignoring a moderator is not a good way of dealing with them
a moderator that needs to chat with a user ignoring them is not notified that they are not being read
a moderator who is ignored will need to escalate to a mod message in order to communicate with a user
I don't really see a valid case for keeping the status quo, and I see many drawbacks both for regular users and moderators.
I think we should make mod chat messages always visible. What say you?
† Chat moderators are users shown in blue color in chat. They are generally mods or employees, but there are a few strange exceptions. For example a mod on any network site except SO or meta.SE will be blue in all the network chats, but not on the SO chat or meta.SE chat.