Cinematic Initiative
The default initiative system is rather rigid, repetitive, illogical, and cause for battles that only take seconds ingame, but hours in real time. This system proposes an alternative method that tackles or solves all of those issues.
The basic concept is a system of consecutive scenes, happening one after another, just like you'd see scenes in a movie. Characters can either be actively part of a scene, or be made part of a scene by another character's actions, and otherwise are assumed to do something reasonably smart that doesn't have any meaningful consequence to the encounter however.
Basics
Initiative is rolled per round. A round is 1 minute long, and subdivided into 6 segments of 10 seconds each.
Player Initiative
Players determine initiative by rolling 2d6. Each rolled number corresponds to a segment in which they will be able to act.
If a character has any kind of bonus (such as advantage) to their initiative roll, the player doesn't apply it. Instead, only for the first round of a fight, those characters get an extra initiative die set to the highest number, effectively allowing them to act first as well as an additional time.
Enemy Initiative
Regular enemies by default roll one die per round.
Boss enemies can roll additional die at the DM's discretion, for example to indicate a particularily agile enemy. An example for a generally balanced boss may be to roll two dice per default, and adding an additional die for each player above four that partake in that fight.
Playing Initiative
Once initiative is rolled, the DM cycles through the individual segments, with the highest roll on the dice corresponding to the first segment being played.
All characters and enemies acting during a certain segment act at the same time. This means that characters can attempt checks together, as well as attempt to interrupt someone else's action.
Whenever no character acts in a turn, the DM can use that timeframe for nonplayer activity, such as enemies regrouping, or even executing their turns.
At the end of a round of initiative, players roll 2d6 again, and a new round starts.