The issue is the dock itself which may need drivers.
Looking online it seems to have drivers for Windows. It doesn't look like HP provide driver for Linux for this.
The problem is to do with display over USB. If you plug a display device directly into a USB-c port in your notebook then it should work as USB-C supports display port directly (display port alternative mode) and this is in the Linux kernel.
However if you plug in a USB hub, then you're dependent on the hub device which is running as a USB device. If it has very good hardware then it might effectively be running more true USB-c ports but it could also (and kuch more likely) rely on drivers like DisplayLink to allow it to run a display over USB.
DisplayLink is proprietary and not in the Linux kernel to my knowledge. You could look to enable DisplayLink drivers in your Arch set up. There is a guide on the Wiki but it is described as experimental. There are official Display Link drivers for Linux (from Synaptics) but only officially released for Ubuntu, so support is patchy.