Better to save blue gems for next playthrough if i want to buy bows with 4 mod slots?

nope. They do not carry over.

They don't carry over but there will be a merchant in Meridian that sells you bluegleam. There is a limit on how much you can buy but it will give you a headstart. To afford them, you need lenses from big animals like behemoth, rockbreaker and thunderjaw.

Originally posted by MadBone12:

They don't carry over but there will be a merchant in Meridian that sells you bluegleam. There is a limit on how much you can buy but it will give you a headstart. To afford them, you need lenses from big animals like behemoth, rockbreaker and thunderjaw.

The Mystery Boxes Merchant is always available regardless of NG+ and amongst his stock of boxes are four Bluegleam Boxes that each contain a single Bluegleam. To purchase all four of his Bluegleam Boxes you require a Behemoth Lens, a Rockbreaker Lens, a Thunderjaw Lens and a Stormbird Lens (one lens per box). For more information about this merchant:
https://horizon.fandom.com/wiki/Mystery_Boxes_Merchant

As already stated, Bluegleam is like a quest item and as such will not carry over to NG+. If you do have any Bluegleam in your inventory when you start NG+ it's automatically removed. For more information about Bluegleam:
https://horizon.fandom.com/wiki/Bluegleam
Might be worth pointing out that there's exactly 100 Bluegleam that you can collect in every single playthrough, which is also the exact amount required to purchase the 3 Master Outfits, 3 Banuk Bows and 5 Treasure Boxes from Bluegleam Merchants.

In other words, there's no point in saving any Bluegleam, just spend it all on acquiring everything you can. You need 42 Bluegleam to buy all 3 Banuk Bows, another 48 Bluegleam to buy all 3 Master Outfits and finally another 10 Bluegleam to buy all 5 Treasure Boxes. The Banuk Bows and Master Outfits carry over to NG+, so it's better to acquire those to give yourself better gear to start NG+ with, rather than not buying them at all.