marco_doz: I spent two hours trying to get it running. This thread was the first result in google but had no solutions. This is how I got it to run, I hope it helps someone in the future.

1) Install the game with the GOG installer
2) In the installation folder, locate and make a copy of both files in VtMB/Unofficial_Patch/dlls/ (vampire.dll and vampire.dll.12)
3) Optionally, copy the Unofficial_Patch folder and rename it Companion_Mod
4) Extract the vampire folder from compmod_1.0.zip into either the Unofficial_Patch or your Companion_Mod folder (you don't need the other files from the companion mod)
5) Extract the vampire folder from compmodv1.4Patch.zip into either the Unofficial_Patch or your Companion_Mod folder
6) Overwrite the files in the dlls folder with the ones you copied earlier (THIS IS THE CRUCIAL STEP)
7) Right click on vampire.exe -> Properties -> Compatibility _> check "Run this program as an administrator"
8) If you created a Companion_Mod folder, edit the properties of the VTM Bloodlines shortcut and change in Target the "Unofficial_Patch" to "Companion_Mod" (or whatever you called your folder)
9) Launch the game through the shortcut

This worked for me and I hope it works for you.

tl;dr: You need the vampire.dlls from a recent Unofficial Patch to get companion mod to run on recent systems.

ludgerarnold: The problem is the mod loader. Comp mod must be installed in the /vampire directory and doesn't work if installed in its own mod directory.
So you should create a second bloodlines install. It should work if you just make a copy of the whole bloodlines directory. Then remove the /unofficial_patch folder and any other mod folders you might have from the new instance. Now you install comp mod into the /vampire folder of your second bloodlines. I'm not sure if it comes with an installer. In that case use the installer, just make sure you install in the right bloodlines copy. If there is no installer or if you already have it unzipped, merge the two /vampire folders.
Now to make sure it will run on modern systems, copy the contents (dll files only, not those with additional extensions) from both the original /vampire/dlls and /vampire/cl_dlls.
Last step, create a launcher or start menu entry.