Multiple monitors, multiple workspaces?

If I understand the question correctly, you would like to be able to switch workspaces independently for each screen, so that for example the left screen could be on workspace 1 and the right screen on workspace 3. This is supported by some window managers, most notably XMonad, and I personally find it extremely convenient in a dual-head setup. Moreover, it's perfectly possible to integrate XMonad with GNOME without sacrificing usability.

On my home machine I currently have a much simpler setup, with the default window manager. I like to have a music player visible on one of the screens at all times, so I just pinned the player to the visible workplace in order to emulate a “separate workspace”. This doesn't really solve a problem, but might be helpful anyway.

It is possible to archive something similar using nVidia (prop) driver (Ge-Force graphics cards). It is called "TwinView". I only tried it once. You may want to read more here: http://ubuntuforums.org/showthread.php?t=221174

I was not able to achieve the same result using an ATI graphics card.

I guess what we (or at least I) really want is to get the top & bottom panels on both monitors and have the bottom panel's application tabs only on the corresponding monitor.

Looks like it's not there yet but we can vote for it on ubuntu brainstorm.

In Gnome

sudo apt install gnome-tweak-tool
sudo gnome-tweaks

in the workspaces tab under "display handling" select "workspaces span displays" which means all the screens to act together. (when you switch on one, others will switch to that workspace as well)

you can select whether all the screens act together or individually to switch between workspaces press ALT + CTRL +UP_ARROW/DOWN_ARROW.

This guy has a working xorg.conf for multiple cards multiple desktops using a displaylink usb adapter: http://mulchman.org/blog/?tag=displaylink