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If I had to come up with a partner 120hp row to permanently sit on the Matriarch it would look something like this. These modules tend to get the most mileage with the synth.

The rear panel module is very useful and is so cheap that something equivalent should have been included with the synth.
She desperately needs more mults and attenuators/attenuverters.
Chaos/random CV (I use Wogglebug and Sloths).
More LFOs or envelopes and a VCA/mixer or two to help with those.
Ring mod if you like it, imo much better with something external to free up the built-in modules.
A high pass filter so you can run mixer -> hpf -> stereo lpf.
Your favorite stereo effect or a pedal interface module like strymon's AA.1. Patching from the filter -> your effect -> back into the matriarch's delay is a dream for example.
Lots of options for this since the Matriarch feels so "finished" in most ways to me, really depends on what you are doing with it and where you want to go. A Pam's might be an easy suggestion for some people but I don't really sync up that much modulation with the Moog in practice. It does so much work on its own that I find myself mostly using "the more basic stuff" to augment it.