You can upgrade Tier 4 pets to Tier 5 with a Royal Plume, and gain the ability to appoint an Alpha Pet. What this does is increase your pet's talent level by one and make the other pets outside loot faster.
The following are the items required for making a Royal Plume for upgrading your T4 pet to a T5 pet:
- Magical Shard x40
- 600 million silver for Growth Reagent x10
- Fine Lightweight Plume x800
Magical Shard can be obtained from heating any Black Magic Crystal.
Growth Reagent can be bought from old moon vendors for 60mil a piece
Fine Lightweight Plume can be obtained by Filtering: Lightweight Plume or Soft Ferri Feather, or you could just buy it from a vendor in Calpheon City Lavientia Batian.
Once you obtain all the needed mats for a T5 pet, you can find the quest chain in your Quest Window (O) under Suggested Tab - [ADV Support][Lv. 60] Appoint Your Alpha Pet, The Best Companion
Once you complete this quest chain, you can use Simple Alchemy on the above mats to obtain a Royal Plume. You can then speak to Obi Bellen in Old Wisdom Tree and upgrade your T4 pet to a T5 pet, you can then set that pet as an alpha pet. Note that you can have only one pet out as an alpha pet, no matter how many T5 pets are out, only the Alpha has the stats of a T5 pet while the other T5s have the same stats as a T4 while not in alpha state. Here are the stats for pets' talents
Talents like Skill/Life/Combat exp are not as worth it since its just 1% increase to a stat that gets soft capped quite early. More better talents people generally go for are Item Drop Rate from Arctic Fox or Max HP from Dragons.
Hope is this useful to any new adventurers, or returning adventurers who may have missed the update or are not aware of this method for upgrading your pet to T5.