I see there is a reference or two missing from my initial post :)
I changed the z-index value in the CSS of the Cookie Law Info plugin not in the theme's CSS.
The site mitja.splet.arnes.si already has the fixed plugin, so the cookie bar stays on top.
I have screenshots of before and after for another site.
http://mitja.splet.arnes.si/2014/10/24/cookie-law-info-plugin
It seemed like a good idea to raise the z-index value in the plugin itself to make stay on top.
My understanding was that the cookie bar should be shown on top of all other content. Is that not so?
Regards,
Mitja