Nice looking GH4 footage...
878 GH4 Videos on Vimeo
I shoot corporate education and training videos for 99 percent of my work, and occasionally I shoot short narrative films with friends, but it's so much work to create something that doesn't look cheesy. ( friends are not too keen on doing the work, which is often a big limitation )
You might want to wait to take a look at the new Sony X70 which should be out within the next month. It's a compact ENG camera that shoots 10-bit 4:2:2 1080HD, and has a 4K paid upgrade coming around the end of the year. Having a long parfocal zoom lens would be very handy for run'n'gun and live event shooting.
Given your budget I would l look at either the Panasonic GH3 or the Nikon D5300. Both of these cameras are in your price range, they both shoot good video and still photos. The GH3 is better for video work, while the D5300 is a better stills camera. Both of these cameras are in high demand, so you should be able to easily sell either of them down the road should you decide to upgrade to a more advanced camera.
One big bonus for the GH3 is that you can shoot 3+ hours continuously on one battery, where you will get less than half of this time with the D5300, and only in 20 minute segments with the Nikon. ( I've shot a 2.5 hour concert with my GH3 and still had plenty of battery power left )
GH4 4K footage is easily recognizable, especially when down converted to 1080 HD. ( it's 1500 lines of resolution in 1080 HD format versus 800 lines or less with non 4K cameras )
As for the "look" of GH4 footage it all depends on camera settings, lenses used, how the scene was lit, and how it was processed in post. Just like almost any other video camera, you can get a huge range of looks depending on your process. Take a look at Kendy Ty's Canon T2i footage that looks nothing like any other T2i footage because of how he processes his footage in post. You can do the same with any other video camera.