Illustrator always adds 0.5 when I move an object

Hamish,

Or you may need to get rid of alignment to the Pixel Grid.

From CC 2017 on, you can see the options and behaviour here:
https://helpx.adobe.com/illustrator/using/pixel-perfect.html

For earlier versions, you may select everything and then untick Align to Pixel Grid in the Transform palette, and also untick Align New Objects to Pixel Grid in the flyout options. It is crucial that the box is completely unticked, not just a - which also counts as ticked.

It may be noted that even if Align to Pixel Grid is unticked with all objects selected, as long as Align New Objects to Pixel Grid is ticked the creation of new objects will result in those new objects being aligned, even when they are created from existing objects as may happen with Pathfinder operations.

You may avoid document types with that as default (RGB for web and the like) or change the default for such documents.

Apologizing for possible dead links below.

Here is an exquisite screenshot made by Ton in the first post for older versions:
https://forums.adobe.com/message/7841770#7841770

And here is an exquisite screenshot by Isa in post #3:
https://forums.adobe.com/thread/2119525

You may also see the screenshots by Tony in post #3 here:
https://forums.adobe.com/message/9397981#9397981

Try deselecting Use Preview Bounds in Preferences.

Hamish,

Is it also happening in any other/new document, rather than in a specific one?

If only in one document, you may be able to copy the artwork in its entirety/in parts/in part to a new document.

Any Snap to options active in your View menu?