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.
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.
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.
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.
I will use her level 20 stats as reference here.
With Compounding Aether: 855 + 100 = 955
With Timewalkers Pursuit: 855 + ~43 = ~898
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.
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.