I've done a few projects where this probably would have been useful, and never had it. And not's not like I can't order one from Amazon for under $40 US, but some times you just want to design your own to be able to use.
What is a height gauge used for? Obviously, it's to measure or gauge the height of something, but it's a bit more complicated than that. Let's consider a D cell battery. You can look up the specs on how large that battery is supposed to be, and you can get calipers, or a micrometer and get precision measurements. There are situations where that's not as easy. Perhaps you want to see if your print bed is level with the path of your print head. Presumably your print head is riding on some variety of rail, whether that's v-grove 2020 extrusions or 8mm chromed drill rod, or linear rails. You can use the usual options, move the print head nozzle to each screw, dragging the head, or bed around, but perhaps that's not a great idea? How about setting up a height gauge to measure the top of the rail your print head is riding on at the left end of the bed, then compare that with the height of the same feature on the right end of the bed. Likewise for a core-xy printer, verifying that you're y rails are level per the print path.
A more useful feature is if you need to measure a features on an object you are replicating, that you can't, or are not allowed to, move or pick up to measure with a pair of callipers. Let's say it's a fairly fragile desk lamp. or a finial for a lamp. You need to know what the heights of arcs are where they transition, or are at minimum or maximum radius. A 'mundane' use I've run into is measuring the height of features on the surface of a soft drink can that I'm going to design or print up cozies for.
In fact one of these has been on my 'to make' list for a very long time. And I figure others are thinking about it as well.
This comes in 2 parts, and needs 2 M3 Socket Head screws. Obviously you can remix it to require more options, and there may be good reasons for that. (A single screw acts as a pivot, which can introduce error into your metrology for example, but I'm already using suspect tools, the error a 1 degree tilt is going to introduce is tolerable. Over a 200 mm span the error introduced is less than 4 mm, I can live with that, if you can't feel free to make changes.)
Print the base with either broad side down/up. If you trust your printer's Ironing' option, this may provide a better surface for the bottom of the base. My intent is for the screw hole in the base to be the front of the base, with the knock-out extension of the base being on the right.
I recommend printing the Index Pointer upside down. The very 'top' surface while printing in this case should be the bottom of the pointer, which will be a very small surface. Ironing this may or may not give you better results.
Assembly: The Index/Pointer should go onto the caliper first. The screw hole goes to the back, and a screw through this should be securing the piece to the jaw physically attached to the display. I'd advise holding off on fastening that at this point, mostly because we'll want to do alignment in a bit. If you do secure it, I'd advise leaving it lose enough to allow you to make corrections later (I.e. don't epoxy it in place just yet.)
Now the base. The end of the jaws should fit in such a way that the screw hole allows you to screw down onto the main thickness of the moving arm. Use the face or a machinist square to insure that the movement is vertical . The moving jaw should (mostly) fit into the base. The pointer will be on the 'left' side of the assembly and that should be the narrower end of the base.
Zeroing:
Once you have the assembly loosely built, I advise going to whatever you're using as a surface plate. Make sure that the calipers are aligned within the base, and secure that. Now close the caliper jaws, and move the end of the pointer to the surface plate. It is not absolutely critical that this be 'level' on the caliper gauge, but having it 'secure' is a good idea.
Now, with the pointer on your surface plate, and everything tightened down (try not to strip those screw holes) turn on your caliper/Height Gauge, select the units you are going to be measuring in, and press the 'Zero' button to zero the calipers. You should now be able to go to whatever you are measuring (If it's not already on your surface plate) and take height measurements. If you have a spare 18650 battery you can make sure that when it's on it's side, it measures 18 mm in diameter, and on end it should be about 65 mm long. (the 0 means that it's a cylindrical battery, not that the battery should measure 65.0 mm long, though that's often the case as well.) Diameter, height or length of the battery may be variable based on the wrap, or whether a battery protection pad has been added to the battery.
Enjoy.
Disclaimer, I've had mixed results with Pittsburgh Calipers, they are relatively inexpensive, but occasionally you end up with battery eaters, where you give it a battery, and 2 weeks later when you go to use it a second time you get to give it another battery. This set tends to last months on a battery, and is my preferred caliper, but I recognize it's not going to give me Mitutoyo results. I also have more than one pair of different plastic or carbon fiber digital calipers, again, variable results. These Pittsburgh calipers will present 3 orders of magnitude accuracy below the decimal, where the cheap carbon fiber ones give me only 2. I don't run into that being a significant issue, but I'd note that if you're looking for greater accuracy, you may be better off buying the Shars or Mitutoyo height gauge.
Don't rely on the numbers displayed in the images here, Between some measurements, the 'zero' button got pressed. Great if you need to work out how large a given feature is, not great if yo're trying to get the height of that feature.
The adapter here is small enough that it can live in my tool box not attached to my calipers until I need that capability.