Intersting, thanks Clark. I do have setpoint installed, but it's not set to manage any windows. I also have Catalyst Control Centre (for my AMD graphics card) which will mess about with windows too. But, again, that's set to not do anything.
To track this down, I started up SE and moved the main window to my second monitor then floated a window and moved it to my primary monitor. I wanted to see if the floated window pops up in the center of my primary monitor, or the center of the SE window. On closing and restarting SE the floated window opened where it was left!
I then tried a few other combinations, including not just my original problem but also the way in which I floated the windows, closed SE, etc. The floated window now acts as it should. Perversely, that's more than a little bit annoying.
If it does it again I'll bear your comments in mind and see if I can track down what's doing this.