Companion + companion matchmaking (cross games)

Non-canon? Here we go!

Bull and Fenris would be a great pairing. 😎 They've both had traumatic experiences on Seheron, they're both familiar with Tevinter and the Qun, and they're both antsy about magic. They're both warriors, Bull likes "pretty" and glowy Fenris is definitely that, and Fenris needs someone who accepts him but doesn't treat him like glass. PERFECT.

Bethany and Cullen. Like Bethany's flirtationship with Sebastian only without all the drama. They're both devout Andrastians, people love to pair Cullen with Circle mages AND he was possibly infatuated with her cousin if you go the female Warden Amell route. Bethany's a gentle soul who's always wanted a normal life; Cullen's romanced epilogue involves marrying his beloved and introducing her to his family in the countryside.

Merrill and Solas. Merrill is fascinated by hahren, and Solas is amazed that there exists one Dalish elf as woke as he! But Merrill has to deal with Baldy's jabs about Dalish ignorance and she would brokenly beg him not to go through with his world-destroying plot. It's a tragic romance.

Sera and Marian Hawke. Age difference aside, purple Hawke and Sera would get on like white on rice. They would make Aveline's life hell and be really, really, ridiculously good looking in the process. Kirkwall would burn, but it's not Anders' fault this time.

I CAN'T BELIEVE I FORGOT THE BEST ONE.

Josephine/Isabela. Isabela is a duelist, that's literally her thing, she could duel Otranto for Josie's hand! Josephine is a romantic who dreams of adventuring and sailing and getting swept off her feet. Plus the Montilyets have boats n shit. They would be the prettiest couple of all time, OF ALL TIME. Isabela would be charmed by Josephine and would enjoy making her blush. Yes pls.

Also Josephine/Zevran. They're both Antivan, she's fending off assassins and that's kind of his shtick now, he would 100% do that sweeping bow/kiss your hands thing.

OMG yes, to both! 

Sera and Isabela.
Leliana and Sebastian.
Cassandra and Sebastian.
Blackwall and Aveline.
Cullen and Aveline.
Iron Bull and Zevran.

I... I'm surprised by how much I like this.

Speaking of cross-game interaction, here's some spliced up Fenris and Leliana banter:

I headcannon that they form a pair if Bethany goes to the circle. Don't even care.
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I'm one of those that ship Alistair and Morrigan. In my first playthoughs Alistair remains warden just so I could see their reunion at Skyhold. And then I keep him in the fade because I don't have the courage to kill my protagonists.

Edit: Because they're also a bit obvious, I like Carver with Merrill too.

Speaking of the devil, Merrill with Fenris would be the worst pairing ever.

Good:

Blackwall and Merrill - both running from something while trying to atone/make better. I think they'd end up making each other better people.

Bad:

Varric and Velanna - joker coupled with the least humourous party member ever. His hidden sadness wouldn't help either.

Bad, but so, so good: Isabela and Bull.....

That actually happened, in The Darkspawn Chronicles!

(Banging Leliana is one of the few things Alistair apparently did in that universe that I can actually imagine him doing.)

I can't see Fenris going for a qunari, though.

It would be a strictly non-romantic pairing, but I could see Oghren and Sera getting along famously. Initially, there'd be a little friction, but they'd eventually bond over a love of crude humour. He'd be pretty cool with the whole Red Jenny thing too.

He wanted to join the Qun with Anders, proof:

Anders: Maybe you should join the Qun, Fenris.
Fenris: We could go together. I'd be happy to sew your mouth shut.

It's just a side what if story but in the DAO dlc Dark Chronicles Alistair and Leliana are actually a couple.

I'd go all happy tear-eyed if I get to see Morrigan and Iron Bull ride into the sunset together. Far, far away... Or sail away on a ship captained by Isabela. Yeah, that's a perfect composite affair there. Morrigan. Iron Bull. Isabela. Ship. Sunset. Awww....