How to hide "Workspace switcher" icon?

right click on the icon and click "remove from panel" :)

It doesn't work. I don't see popup menu. Do you see?

Nope, there's no such option. Can't even move the icons. Currently, Unity is pretty much useless with a dual monitor setup.

I fail to see how it is "pretty much useless"... it displays and works fine.
Not completely optimal, sure, but it works better than a lot of other
things. Let's stick to a realistic analysis.

"Works fine"? See bug #711916 and 14 other critical bugs, no fix released yet, excluding incomplete ones. [1]

It doesn't work fine, it works acceptable. It lacks the most basic features people expect to see, or even commodities:
- removing (some) icons
- organising icons
- having the option to change settings without having to install ccsm
- proper dual monitor support

Honestly, it needs a lot of work. People are just not going to buy it like this. It's like a car without a gear shift. Sure, you can drive, but not like you expect you can. Without these basic features, it is pretty much useless.

[1] https://bugs.launchpad.net/unity/+bugs?field.searchtext=&orderby=-importance&search=Search&field.status%3Alist=NEW&field.status%3Alist=CONFIRMED&field.status%3Alist=TRIAGED&field.status%3Alist=INPROGRESS&field.status%3Alist=FIXCOMMITTED&field.importance%3Alist=CRITICAL&assignee_option=any&field.assignee=&field.bug_reporter=&field.bug_supervisor=&field.bug_commenter=&field.subscriber=&field.tag=&field.tags_combinator=ANY&field.has_cve.used=&field.omit_dupes.used=&field.omit_dupes=on&field.affects_me.used=&field.has_patch.used=&field.has_branches.used=&field.has_branches=on&field.has_no_branches.used=&field.has_no_branches=on&field.has_blueprints.used=&field.has_blueprints=on&field.has_no_blueprints.used=&field.has_no_blueprints=on

I too would like to get rid of this button. I'm using the compiz cube and switch desktops via the viewport switcher plugin. So, the button on the launcher is just wasted space.

I hate when the U.I. designers are adamant about locking you into one way of doing things when they haven't thought of all use cases.

I haven't found a way to do it yet... still searching...

A workaround:

http://askubuntu.com/questions/41730/is-there-a-way-to-disable-the-workspace-switcher-in-unity-launcher

That workaround doesn't work for those of us using non-wall-implementing multiple desktop plugins.