One of the problems I've currently got with the Chase rules is that, because only pursuers make initiative rolls, the fleeing character never triggers a complication. The obvious solution is to make the fleeing character roll as well, but that would make it inconsistent with regular initiative.
However if everyone rolled initiative in regular combat, it would add even more rolls, and the more I think about this issue, the more I feel that initiative is already overly intrusive. Savage Worlds manages to get away with calculating initiative every round, perhaps because the cards feel like such a different game mechanic to trait checks, but in Swift d12 the extra roll can become annoying.
The main design goal for the original initiative system was to ensure that one side didn't always get to act before the other (because there are often large numbers of combatants on the battlefield, and one side could cripple the other if they all got to attack first). Any alternative system would still need to address that original design goal.
One interesting initiative system I've read about is the one in Shadow of the Demon Lord, where characters can choose to take either a "fast turn" or a "slow turn". Players act before NPCs within each turn, and those in the fast turn take one action while those in the slow turn take two actions, so acting first comes at a price. This is a quick and clever mechanic, and feedback from players seems to be generally positive. While the solution wouldn't work as written in Swift d12, it's given me inspiration for something along similar lines.
Alternative Initiative Proposal
At the beginning of each round, the Game Master asks if any players wish to "rush" their actions, and those who rush automatically act first. After that, the Game Master may decide to make any NPCs rush their own actions, and they act next. Then the remaining players get to act, followed by the remaining NPCs.
However characters can only rush if they're not Staggered, and they become Staggered when they rush. Of course they can still use the recover action to immediately remove the Staggered condition as usual, but recovering is a simple action, so it'll cost them an action die.
Interaction with Other Mechanics
The Lightning-Quick Feat already allows characters to spend a Karma Point to take a simple action as a free action, so with the alternative initiative they could "rush" and then spend a Karma Point to recover.
In a surprise situation, the surprised characters always start combat Staggered, and this would prevent them from rushing.
The All-Out Attack maneuver gives characters +2 to attack and damage, but they also become Staggered. What I wanted to avoid in the original initiative system was a situation whereby players could choose to act last with an All-Out Attack, then act first in the next round and recover before anyone could take advantage of them being Staggered. But as Staggered characters cannot rush, this wouldn't be an issue with the proposed rule.