In Eclipse, can I have multiple Console views at once, each showing a different Console?

Yes,

• located near your console tab should be a button "Open Console".
• If you click this button one of your options should be "New Console View".

You'll now have 2 console views.

• One of your other buttons near your console tab is "Display Selected Console". When you choose this option you can select from any of your running applications.
• Just select the tab, select which application you want it to watch, and repeat for the other tab.

You can then move your 2 console views to wherever you want independently of each other.

I'm using Eclipse Helios Release with build ID: 20100617-1415.

The best thing you can do is the following.

Window > New Window. 

That will create another eclipse window like you currently have. Run the desired application, switch the console on the new window to display the desired console output with open console button on the console view.

A bit clunky but the only way to get multiple consoles at the same time. If your new windows doesn't have console open use

Window > Show View > Console

Goodluck