While you can shoot with whatever kind of anamorphic lens you want on the Panasonic GH4 assuming it covers the sensor, many of them will not give you a proper aspect ratio without cropping a ton.
1.3X anamorphics have been popular because they take 16:9 and make it the correct wider aspect ratio, but the anamorphic effect isn't as strong as 2X anamorphics. The new V2.0 Panasonic firmware update has added 4:3, 3:2, and 1:1 aspect ratios that use more of the sensor height, so you can actually get closer to the correct aspect ratio using 2X anamorphics without significant cropping (or an insanely wide aspect ratio) that comes from using 2X on a 16:9 sensor. There is a pretty decent catch at the moment, however.
Here's some footage shot with this new mode from Neumann Films:
Video is no longer available: www.youtube.com/watch?v=nuZZnW8R8Vg
Here's Luke Neumann to explain how he shot the footage above and how you can get the most out of the GH4:
Video is no longer available: www.youtube.com/watch?v=j_OpGudiCbw
While he mentions in the video 30fps is only frame rate, you can also select 25fps as well for PAL. Unfortunately, neither of those are the frame rate many of us really want, as 24fps would be preferable at least for filmmaking purposes here in the US. Hopefully at some point Panasonic adds 24fps, as it would really make this the cheapest and most useful camera for anamorphic shooting with all sorts of different lenses.
He's made available some test footage, and a LUT that he's been using. You can find both of those here.
Besides adding 4K 23.98 at 100mbps in MP4, here's what's possible with the new V2.0 firmware update: