Hi,
I must disagree with the post that says you can achieve haze reduction and rarely need polarizers due to what you can accomplish in post processing. I am a firm believer in maximizing quality at the moment of capture. With the right filters, literally, five seconds in the field can save you five hours behind the computer screen.
You can get the appearance of haze reduction but are taking quality out of the image. Better to eliminate the haze optically at the time of capture. It is true that a polarizer also takes out haze, more noticeably than the UV filter, but the UV can add still more reduction, especially at high altitudes or in aerial photography where UV rays are more intense.
Also, you really cannot duplicate the polarizer effect in post processing, especially the glare reduction which reveals the existing natural color in a way different from pumping up saturation digitally, and also reveals detail not able to be seen otherwise. The darkening effect on skies helps the dynamic range of the image, in extreme situations. I use polarizers 90% of the time in outdoor shooting, even on overcast days, where they still can have a noticeable effect (they cut glare on surfaces even in overcast conditions, and really intensify rainbows!).
As for multicoating, it is true that the use of a lens hood is the best way to beat flare (and also protects your lens), but, with extreme zooms and "petal" hoods a lot of stray light (including oblique sunlight) can still reach the lens surface. For these cameras especially multicoating is still valuable and well worth the cost. I would say that if I had to choose an uncoated linear polarizer or a coated circular one, I would go with the circular for the coating benefits alone. I would sacrifice the coating only if you will hardly ever use the filter or if the extra money is completely prohibitive. For me, the polarizer is THE essential accessory for outdoor work (my specialty) which you will use for may years, and I get the best I can afford. If you invest in the Leica optics of the FZ50 then I would suggest the extra $20 for a multi-coated filter.
Finally, I still shoot film, very happily, with my EOS I-V and Contax 645 medium format system. All my film cameras since my Nikon F3 (am I dating myself? -- ouch!) recommended circular polarizers and the camera manufacturers' polarizers (Nikon and Canon) were always circular. With this infol, I never tried a linear one on these--perhaps paying too much all this time

!
Good Shooting!
Guy