The thing that will help the most is to match your master pedestal (black level) and to grade as close as possible to what you're seeing from the Alexa in-camera. What our colourists are seeing is not so dependent on the exact camera settings as it is on how close you can get in-cam so the adjustments they make are minimal. It will match VERY closely (almost indistinguishably) with care.

So what I'd do, if I were in your situation, is try to get some close-to-final-grade Alexa footage from the scenes you're trying to match, take the GH4 out, and dial that in. I'd recommend starting with either CineV -5 -5 -5 0 H/S curve lower contrast or Cine D 0 0 0 0, H/S curve flat, 4k (and downsample in post to gain some colour and reduce banding). For the MP setting, put the lens cap on an Alexa in the rental shop and grab a few seconds of black, do the same with the GH4 and adjust the MP to match the black levels in the final footage. Once that's done, you'll end up with damn near the same range after grading and the shadows will look great. Watch your highlights when shooting and you should be pretty close.

Remember, don't trust the GH4's back screen, but use an actual calibrated monitor like a Panasonic field monitor or similar. The GH4 output should be close to the Alexa footage AFTER the Rec709 LUT is applied (assuming you shot LogC), or with near-final grading, if you can get it.

Resolve 11 and the parade are your friends for matching black level and general tonal ranges, use the waveform when adjusting hue, if necessary. Do colour as minimally as possible in camera - with the exception of setting your white balance properly and making sure that you don't have awkward skin tones in-cam.

Great advice, thank you!

There's nothing wrong with a GH4 - it can very easily match with the Alexa when handled properly.

The 4K resolution will actually compare better - the Alexa is roughly 2.8k resolution versus the BMPCC's 1080p, and the depth of field is in the GH4's favour with less expensive lenses. Colour isn't the end-all for matching, and while the BMPCC is indeed a great camera, in this case they'll do just as well if not better with the GH4.

As for codecs, you really don't need RAW or ProRes when your scene is lit right and your codec is good enough, especially if you are downscaling from a much higher resolution which will reduce the noise, macro blocking, and other potential artifacts from compression. And if he goes to a Ninja Star for ProRes, the codecs and resolutions are equal and you still gain the depth of field, lens flexibility, and overall rig handy-ness the GH4 can offer, without losing on colour.

So, overall, the GH4 is at least "good enough" here, and if I were given the choice between the BMPCC and the GH4 in this case, all else (lenses, lighting, etc) being equal, I'd probably take the GH4 too, unless my tests showed a significant advantage elsewhere.

Of course, I'd put the majority of my effort into light control and the rest of my money into glass, if it were me, and not really care two whits what I'm shooting it on as long as it's sufficiently capable.

If intercut with existing footage in the same scene, matching the lighting will be the most important aspect of your work. If shooting entire new scenes, matching the look (including the lighting) style of camera movement & operating will be more important than GH4 vs BMCC (IMHO).

@HenryO - Apologies, that wasn't aimed directly at you, but rather just a general comment for the OP. Of course the BMPCC is a very capable cam as well, but I don't find that the GH4 (which I own) is any harder to match than the BMPCC (whose footage I have used) to an Alexa (which I have been DIT for and handled prelim colour on). I just wanted to set this often misunderstood record straight.

As CFreak and mrbill are also saying: for both BMPCC and GH4 (and any other camera without professional-grade-log 10-bit ProRes or RAW) the absolute most important thing to do is match the lighting, nail your exposure / WB, and copy the shooting style, and of course I would add try to use the SAME glass as the Alexa did.

Everything else is really academic.

To elaborate on what @mrbill is saying, Master pedestal is specifically there only to match black levels. It does a good job at that. You'll find that C300, Alexa, F55, RED, etc all have different base black levels. The only adjustments to MP you should make are to match another cameras black levels which are very useful and as important for overall accuracy as white balance for matching. Panasonic themselves even state clearly this is what that control is for and nothing else.

Shooting black with the lens cap on for instance is a very good way to set this level to match another camera in my experience and it does in fact help to match shadow handling.

But it is not an excuse for tweaking for creative purposes or to get the "log look" which is not the proper use for this control.

Great to hear, @discomandavis, I'll keep an eye out. If you learned anything else from your experience feel free to share. Each shoot is different and there's always something to be learned!