Using Gimp, given an input image, I can improve its contrast using Colors > Hue Chroma... by setting Chroma=50 (in a scale between -100 and 100) and leaving Hue=0 and Lightness=0. So it appears I'm doing an HCL transformation.
Is there an equivalent command for Magick?
The following image shows the GIMP effect: Image
Updated Answer
Not sure about this at all. I think you can get pretty close with -modulate if you go into an LCH colourspace, but I have no idea if it will work consistently. I got:
magick cXDv3.jpg -define modulate:colorspace=LCH -modulate 100,150 result.jpg
If that doesn't work, or is not to your liking, read on...
The method below should allow you to replicate any GIMP filter with ImageMagick - as long as it is a pure "point process", I mean one where each pixel's output value is purely derived from its input value and not an "area process" where surrounding pixels contribute - such as blurring or median filtering, for example.
It's called a HALD-CLUT. You would create a HALD-CLUT something like this:
magick hald:16 clut.png
Then take that file (clut.png) into GIMP and apply your GIMP processing on it and save the result as GIMP-H0-C50-L0.png so we know how GIMP affects each colour. You do that just once.
Then you go back to ImageMagick and apply that CLUT to your image:
magick input.png GIMP-H0-C50-L0.png -hald-clut result.png
That gives me this:
and I think you'll agree the left side looks pretty similar to the right side of your input image.
Original Answer
I don't know what that command does in GIMP, but you can convert to HCL colourspace in ImageMagick and select the Chroma channel for modification like this:
magick INPUT.PNG -colorspace HCL -channel G ...
You then want to do something ? to affect the Chroma channel, so try -auto-level for now, and then return to sRGB colourspace and save:
magick INPUT.PNG -colorspace HCL -channel G -auto-level +channel -colorspace sRGB RESULT.PNG
Then you need to provide more clues or experiment more with what that command does in GIMP - or provide examples.