8/8 FREE, PROTECTED
We've got four Inquisitors currently in the Blue Moon universe who alter the deck-building recipes for the people we choose to play with. They are:
Razor-Mind: +5 cards, +5 moons, all cards get the REPLACE icon
Grease-Palm: +10 cards, +15 moons
Weave-Tongue: +15 cards, +10 moons, draw an extra card at the end of each turn
Swift-Fist: +0 cards, +0 moons, attract one dragon at the start of the game
Some will be naturally better-suited to accompany certain people moreso than others. Here is my people by people breakdown of what Inquisitors make good matches, generally speaking.
Hoax - Their versatility means they can be built to accomodate any of the Inquisitors. Swift-Fist might be the worst match of the four, as he is best suited for blitzing decks or decks with a very fine balance that you don't want to muck up by adding too many more cards. The Hoax are neither of these.
Vulca - Their strong Fire values and dependence on powerful Support cards makes me lean towards pairing them with Grease-Palm, who can let them do that but moreso. Weave-Tongue is another good candidate if you find yourself adding many cards with low moon costs (such as Mutants or Hyla).
Mimix - As a blitz deck, Swift-Fist is a good match. The idea is that by hitting hard and burning through your deck quickly you either get an instant win quicker or don't give the opponent time to dig out of the 1-dragon hole that Swift-Fist starts them in. Weave-Tongue is a good match if you don't want to rush with the Mimix as you can use the extra card slots to bulk up on the boosters and supports that the Mimix are lacking.
Flit - Swift-Fist is also good for the Flit, as their low character count makes one hesitant to add too many extra cards to the deck (unless they too are characters) and the paired boosters benefit from not having too many other cards to prevent them from being in your hand together.
Terrah - Like the Vulca, the Terrah can pack an even heavier punch with the help of Grease-Palm. Another possible route is to focus on making the Storms easier to get together by using the unique ability offered by Razor-Mind.
Khind - Razor-Mind and Weave-Tongue are both fast friends of the Khind, as both make finding matching gang members easier. Use Razor-Mind if you want to play a fast Khind deck and Weave-Tongue if you prefer to slow-play them.
Pillar - Like the Hoax, the Pillar are flexible due to their ability to throw roadblocks in front of their opponents in the form of Caterpillars and to change their own elemental powers thanks to their numerous Potions. As such, any of the Inquisitors could be used effectively. Also like the Hoax, Swift-Fist is perhaps the least good fit of the four.
Aqua - Yet another deck that doesn't seem to have a clear preference, though Razor-Mind is a natural fit to the Aqua's theme of letting the cards flow through your hand with little cause for concern, for you know that the Water of Immortality is waiting to bring them all back.
This is by no means meant to be definitive or exhaustive, but merely a starting off point for discussion. Though some threads have been made concerning individual decks that have been designed, I think talking about some more general aspects of deck-construction would be useful and this is as good a place to start as any.