Open two copies of the same file side-by-side in Photoshop

Looks like this feature does exist. It is called "New Window for". You can find it under:

Window > Arrange > New Window For [Image File Name]

Here is a tutorial: Dual View Photo Editing In Photoshop

And from the Adobe site: http://help.adobe.com/en_US/photoshop/cs/using/WSfd1234e1c4b69f30ea53e41001031ab64-74d5a.html

Go to Window > History.

At the bottom right side of the panel, click on the first button that says "Create new document from current state", and you'll have a duplicate of your document.

If you mean having a copy without messing the original file, go to Image → Duplicate.