Which first-tier talent for Chromie offers the biggest damage boost?

For the very first talent you can pick between 4 choices.

I was wondering what is the difference between Compounding Aether, Deep Breathing, and Timewalker's Pursuit in terms of damage.

Is it better to specialize into Dragon's Breath or Sand Blast for the corresponding damage boost?

Or should I use Timewalker's Pursuit for a flat 5% ability power?

Obviously there are various pros and cons to each choice with respect to other talents. I am trying to understand how effective the 5% ability power bonus is.

Solution 1:

Hotslogs Stats

The first address for questions like this is HotsLogs. Filtering for Diamond and Master League gives you the best representation of the currently played talents. There you see that Compounding Aether and Deep Breathing are picked the most with Deep Breathing having a slightly higher winrate. However, this doesn't help with understanding why that is, so I will just breakdown the 4 choices here shortly.

Compounding Aether

After hitting 25 heroes with your Q you it deals an additional 100 damage. This is the highest damage you can get out of the level 1 talents since your Q has a very low cooldown. It is also fairly easy to get those stacks going since the cooldown is so low.

Deep Beathing

This talent is very similar to the previous one. The difference is that you already get bonus damage while the stacking is still going on, which is definetly a nice benefit together with the increases sigh radius after you finished the quest. However, it takes some more time to complete the quest, since the cooldown is quite long. Also it is not that easy to hit your W without some kind of root/stun setup. Getting the W talents on 4 and 7 helps a lot with this though.
While Compounding Aether increases your sustained damage, this talent is increasing your burst which is often more valuable. Also it provides you with a bit more waveclear. If you team is lacking waveclear, you should definetly go for a W-build.

Timewalkers Pursuit

This talent is a slightly modified version of Conjurer's Pursuit. The 5% ability power is nice, but it comes to play way too late. Chromie is not a hero that can pick globes up very easily. Basically the only way to get globes is when a teammate picks them up for you. Getting them yourself is jsut exposing you to ganks. So just will need a lot of time to complete the quest, while the other two quest talents are finished in a much shorter time.
Also mana is not a big problem as Chromie, so overall this talent is not worth the struggle.

Peer Into the Future

Basically a weaker version of Clairvoyance. There is no real reason to get this talent. If you need reveal vs Nova or Zeratul, you just draft Tassadar, Brightwing or Kharazim.

Damage Breakdown

I will use her level 20 stats as reference here.

Sand Blast

Base: 855
With Compounding Aether: 855 + 100 = 955
With Timewalkers Pursuit: 855 + ~43 = ~898

Dragon's Breath

Base: 965
With Deep Breathing: 965 + 200 = 1165
With Timewalkers Pursuit: 965+ ~48= ~1013

So Timewalkers Pursuit actually provides less damage then the other two talent choices. Of course the ability power bonus applies to both her damage abilities rather than just to one, but even then it's equal at best.
And even if it would provide more damage, as already mentioned, it comes to play way after the quest talents which overall lowers it's impact on the game.

Summary

If your team lacks waveclear, always go for a W-build with Deep breathing. Also overall this is the better choice since the burst it provides is more valuable when you do your combos.
But this doesn't mean that you should never go for Compounding Aether, On maps like Cursed Hollow or Towers of Doom a Q-build gains a lot more value, since poke is very strong on this map. You can interrupt curse or altar channels all day long, if you position yourself right.

Solution 2:

The 2017-09-05 patch notes for Chromie changed everything.

Mounting Sand, the (Q) talent, does not add to Chromie's damage directly. Instead, it allows accurate Chromie players to complete the innate (Q) quest more quickly. If you can land Q attacks without missing, it takes 31 hits without the talent or 21 hits with the talent to complete the innate (Q) quest. If you always hit except on a third consecutive attack, it takes 59 attacks to complete the innate (Q) quest with or without the talent.

Deep Breathing, the (W) talent, adds 140 damage to the W skill when the quest is complete.

TimeWalker's Pursuit, the passive talent, gives 5% spell power when the quest is complete. This amounts to 22 damage for Q and 52 damage for W at level 20. It should also be noted that getting to 15 health orbs is a lot easier now that there are two available per wave.

So now the tradeoffs and playstyles are much more obvious.

Do you want to do the most damage possible in the end or to clear minion waves and do a little damage to buildings/mercs? Take Deep Breathing.

Do you want to present a continuous threat and take the mana intensive Ultimate: Shifting Sands? Take TimeWalker's Pursuit.

Are you amazing with skill shots and want to unlock 40% extra damage on Q as quickly as possible? Take Mounting Sand.