The title pretty much says it all. Is there a brand that stands out in your mind that you would suggest, or avoid? I doubt I'd be willing to spend $100 US for a filter. So less money than that, I think.
I have this one and I'm happy with it. Bought it used when I picked up my 12-40. I also have the one from Nisi and Formatt that works with their rectangular filter systems, but when I'm not using grads or ND's, then the Hoya one is the one I go for.
I have used quite a few different polarizing filters (on different systems/lenses):
B+W branded filters (regular, MRC or Kaesemann are all great) - what I like is the brass filter ring (never had a problem with filters getting stuck) and glass is very water/smear/scratch resistant and super easy to clean;
Hoya the more expensive ones are very good (equal in color rendition to B+W). The regular ones have good performance vs price (but no coatings and performance in slightly less compared to the more expensive ones) - I do not like how easy the Hoya filters are to clean I do prefer B+W in this regard;
Formatt Hitech Firecrest - polarization is good and color rendition decent but the glass is hard to clean and the filter gets almost every use stuck on the lens/other filter threads (and as this is a slim CPL it's very difficult to get unstuck);
Cheap generic CPL - polarizing effect decent but color rendition not very accurate. You would be better off with a reputable brand (even least expensive option);
Kase magnetic circular polarizer - very good color rendition and with the benefit of mounting the filter using a magnetic ring on your lens where the polarizer clicks/attach on to (couldn't be easier);
Quick tip: generally "cheaper" circular polarizing filters from a reputable brand perform quite close to the expensive and very expensive ones. No noticeable difference in polarization, color accuracy in the expensive ones is slightly better. The cheaper ones (from reputable brands) do have some pro's 1. they are cheaper 2. most often do block more light (functioning as a 2 stops ND filter as well as polarizer, which can actually be a benefit depending on your use case).
Extra tip: If you plan to stack multiple filters on the 12-40 2.8 (and use it at 12mm) buy slim filters, and try it in a store. I experienced vignetting at 12mm combining a regular Hoya ND filter and a slim Formatt Hitech Firecrest Polarization filter (62mm). I guess with two slim filters it could have been ok, but I'm not completely sure.
You could also buy bigger filters and use step up rings from 62mm to for example 77mm which could eliminate vignetting (at 72mm I still had vignetting and decided to invest in a square filter system).
BTW: Has anyone had any experience with this filter adapter for the Panasonic 7-14 f4.0? I'm thinking that maybe I can get a polarizer for the Panasonic lens and use a step-down ring for the 12-40. Can anyone see any problems pending for this idea?
No experience with the filter adapter but for the 7-14 you should keep in mind that the wide field of view (certainly at 7 mm) can result in uneven polarization.
It's due to the nature of how circular polarizers work as the effect is strongest with directional light at 90 degrees to the sun. If you position the camera with polarizer shooting straight into the sun (0 degrees) you would get no polarization, with 45 degrees you would get maximal 50% polarization and so on. This poses a problem when used on ultra wide-angle lenses.
The Panasonic 7-14mm has an 114-degree angle of view which means that you can never get the whole frame 90 degrees to the sun which can result in dark (polarized) and lighter (less-polarized) areas in your final image. Random example photo see this fstoppers article under point 1
Often you do not see this "effect" very well in the viewfinder or LCD screen, but you will definitely see this in the final image.
Not to advise against using a polarizer on a wide-angle lens per se. Zooming in a bit would reduce/remove this effect and not all use cases it's really a problem (for enhancing colors in woodland for instance). But it would be something to be aware of.