Great idea.
GREAT IDEA for 4-way TEAMS

Related BUG:
I was invited by premium user to play 4-way team together.
After we won first game, I pressed "play" to continue onto the next game but I was matched with another random teammate. This should be fixed.
(This is probably major reason why people are resigning at the start of new matches)
I my opinion all the topics raised here could be integrated nicely

Currently, you have to keep inviting the player for every new match. This will be remedied in future updates.

yeah I agree on this point, think it should be like in bughouse, where u keep playing with your partner until u end the partnership
still I disagree on the premium requirement

Yes, in the future we would like to have partnerships persist. @veni-vidi-vici3, please read https://www.chess.com/clubs/forum/view/premium-membership-to-select-own-player-in-team for more on the premium justification. You've been a great contributor as we've been building 4 Player Chess these past couple of months. I would love your continued support as we move forward.
Please implement:
When choosing to play with random opponent, after the game finishes, you could be presented with option to continue with your teammate to the next game. If both players choose to play together, they will play together in the next game.
If you like this idea
please write so below!