"You stand before the King of Thorns, YOU STAND BEFORE VENGANCE ITSELF!"
Where Amaranth was a physically-oriented boss, the Bearer's attacks are completely magical. The Bearer has a variety of attack patterns. He can shoot a stream of high-arcing purple shots that land on the ground to form lingering, damgaing purple puddles. Along with this, he has multiple bullet behaviors as follows; he can shoot a fan of five slow, but long range shots. Shoots a burst of 8 bolts in all directions, sometimes with straight paths out from the Bearer, some times with curved paths clockwise. Shoots five shots in succession, starting with a small width and medium speed, and increasing in both attributes with each shot. Lastly, he can self-detonate in a medium range, red explosion while teleporting to another location.
By far the easiest strategy for any character is to bring one or two Aged Hourglasses with you into the boss fight. This 'strategy' is almost always possible with every character, because there are 9 floors before the fight that each house multiple item chests- it's very likely that you will find an Aged Hourglass in one of those item chests. The Aged Hourglass nullifies enemy magic, and since the Bearer is a completely magical boss, it essentially nullifis the boss's threat to nonexistence.
Use your Hourglass, run up to the boss and begin attacking it continuously. It's worth noting that even with the Hourglass, it is possible for the boss's fastest moving attacks (the final shot in the five-shot increasing speed succession, and the teleporting energy blast) to hit you before they are nullified if you are extremely close to the boss.
It is still possible to beat the boss without an Aged Hourglass, but it will make the fight more difficult without it.