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Standard initiative rules do far more damage than good. Without Initiative, the sequence goes like this:
1 Roll to attack/
2 └── Roll Damage
That's 2 steps, and if we add initiative, we have 3:
1 Roll initiative
2 Roll to attack
3 └── Roll Damage
For 1 PC + 1 Monster, that's a potential 6 steps! It's a long process just to have a fighter crack a zombie over the head, and we're not even looking at counting modifiers.
"The zombie lumbers towards you."
"I hit it with my axe!"
In this scenario, we want the PC to go first, because zombies attack slowly.
When this happens, the initiative roll was pointless, because it did not affect the outcome in any way.
Initiative also makes no difference here.
Once again, initiative makes no difference, because nobody died. It might make some difference if zombies lost their action after receiving a hit, but they don't.
Finally! The initiative-rolling did indeed make a difference, because if the zombie had one initiative, and then successfully attacked, then it could have dealt Damage.
So in most situations, these Initiative rolls do nothing. If we add a full adventuring party into the mix, initiative works a little differently, but often penalizes the player, as they must decide which action to take before everyone else decides, meaning they act with less information.
A lot of people discard the Initiative system and allow people to attack whenever, as narrative or volume dictates. As long as PCs each receive the same actions per turn, this makes for a very clean solution.
In the case of our zombie-attack, both roll to attack. The dice can still represent the zombie's 'slow' attack with the zombie's low attack score, rather than making a pointless roll.
A largely abandoned method from A,D&D involves each side rolling 1D10, and the side with the highest roll goes first - either all players, or all monsters.
I've never seen this approach, but for a slight improvement on the existing system, we could simply read the attack result twice, so characters with a high attack score go before anyone else. It would push players into declaring attacks immediately, since they need this roll to begin acting.
Alternatively, the initiative roll might be the same as the attack roll, with different modifiers. Of course, in this scenario, players would know their attack roll before declaring they want to attack, but that might not be all bad.
I roll an '8'. With +3 for Initiative, that's an Initiative of '12'.
If I attack with my +2 bonus, I would only hit against AC 10, which isn't much good. Okay, I think I'll cast a spell instead...
All previous alternatives relied on reducing Initiative rolls, but we might tip the scales another way - instead of reducing the initiative mechanics, we can make them more valuable in order to justify all that work.
D&D currently has rules for making multiple attacks in a single round - these require referencing some chart concerning a character's level, which is a bit of a hassle for a GM with 5 NPCs to govern. Instead of the chart, we could just say that attacking 'costs' some amount of Initiative. That way, the Initiative bonus also servers to allow characters to attack lots.
We could add different costs for movement, spell-casting, firing arrows, using Feats, et c. and let people spend Initiative as a resource. We have more work than the standard Initiative system, but by adding options we can allow players to make real decisions with their Initiative.
Since Initiative only applies at times, people could just declare their actions in any order, and we reference Initiative rules only when disagreements arise.
I go for the skeleton - that's an '18', so I smash him with my axe for 10 Damage.
The skeleton tries to stab you before you destroy it. It hits you too, so roll for Initiative! The skeleton rolls '11'.
I want to smash the skeleton, before it stabs Rognar...okay, I rolled '5', so I can't?
Rognar rolls a '10', so I guess I'm getting hit.
...it's a little circumlocutious, and would probably have to resolve like Magic: The Gathering's rules for Instant spells on the stack. But it has the advantage of meaningful decisions; notice that when Rognar's companion went to attack the skeleton, they already knew the skeleton's Initiative was high, making the counter-attack a rather bad decision.
We now have a full resolution:
1. The skeleton stabs Rognar. 2. Rognar destroys the skeleton. 3. Rognar's companion stabs at the (already obliterated) skeleton.
Various sub-rounds could occur as different people choose to react or not react, and Initiative rolls can be discarded once someone misses.