I'd say save your money. I was worried about the same thing, and also how a digital camera does with or without a polarizing filter, as well. I got a filter set, with UV, polarizer, and Sepia, all are a waste of money. The computer inside the camera resets the image so it results in no effect in the LCD. I use all of these filters extensively with my film camera, but digital is so different.
I use a UV filter to protect the glass on all my lenses, but the sky comes out about the same (I can't tell) on my main digital camera with filter, and my back-up digital camera (which has no filter).
Most of what a filter will do on a digital camera can be set by photo software. While you can't darken just the sky by a polarizer, you can adjust contrast and brightness to "clear up" a hazy view. Besides, using a polarizer on a film camera makes the image look sort of phony or weird (to some). I need a clear image, first, since I record rock walls and routes up peaks, as well as the summit views.
Then, I have a Nikon Coolpix with Adobe software.