Dear Mr. Moodle: How do I hide myself from the online users block?

Here’s a question that came in recently to Mr. Moodle:

Dear Mr. Moodle!

Some lecturing staff where I work are requesting the option to change their Moodle status from online to offline as/when it suits them. They state privacy reasons – that if they choose do a lot of Moodle work online at odd hours they would rather this not be public knowledge and also so that they cannot be disturbed by students at certain times.

Can you tell me how I could add this feature to my Moodle installation (ver Moodle 1.9.11) or is it even possible?

I’m using Moodle 1.9.10 for a few projects and can attest that I never thought of these issues (doesn’t necessarily pertain to self-paced courses).  It strikes me that one of the easiest [temporary] way would be to hide the various tools that students can use to be able to monitor who is online.  Namely the online users and people (participants) blocks.  Without these blocks students have fewer ways to view online user status, last access or even see when someone else is online.  And when you’re ready for students to see it, just click the eye and voila: social communication is back (just not between teachers and students).

Is there another way to “log out” or “go invisible” like other chat tools that you know of for Moodle?  If there is share it in the comments!