All of the professions are split in to seven different tiers, from apprentice all the way to illustrious. Here, we're going to be taking a look at the level of Master, which starts at the alchemy level of 285 and goes up to 350. Your character level doesn't necessarily need to be high, but for certain quests, like the specialization quests, you will need to be a certain level in order to complete it.
At the master level of alchemy, you are able to create major and super healing potions, as well as a super mana potion; as mentioned,

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you are then also able to pick up the quest in which you determine your specialization. These specializations are broken into three - potions, elixars, and transmutation - and allow to create extra items for the price of one. For example, if you are a potions master, you may be able to create 5 potions for the same materials it would take to create one potion.
In order to obtain the specializations, you will need to be at a minimum of 68 in level or the recommended level of 70. Transmutation seems to be the easiest to obtain as it does not require you to enter a dungeon, while both potion and elixar masters must dungeons (Tempest Keep and Caverns of Time respectively) to complete their quests. If you're an alchemist with plans on entering either dungeon, go for either one of the masters.
Your character can only hold one specialization at a time, though you can switch by going to the quest giver of the master route you decide. Note that you will also need to pay 150 gold in order to switch.