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Field boss finishes, with the exception of Yeti, are based completely on the number of hits you get on the boss.
If you're a human, dual daggers are the best way to get hits in. If the boss has melee passive defenses, use Final Hit whenever you think you won't get interrupted while using the skill. Using Final Hit is also a good way to get around Giant Lion Roar's stun time.
If you're an elf, shoot a lot with a Short Bow or another fast speed bow while standing on top of the 'center' of the boss. The center of Dragons are at the base of their tail, or a bit back from between their legs.
The hits your summoned pet gets in count for you. Using a pet with fastest attack speed will help a bit.
Ifrit
Quickest way to take Ifrit down solo is to make it Windmill. Best way to do this is to Icebolt with a chaincast Ice Wand or shoot a long range bow until he loads Windmill. It should be obvious when he is loading Windmill since it doesn't cause him to flash white and he'll be standing still. Use a pet to trigger the Windmill to minimize your EXP/Blessing loss.
Neid
Windmill whenever there is a skip in animation.
Giant Alligator
Windmill whenever there is a skip in animation or opens its mouth. You will not be able to attack it when it is starting and performing a 360° tail swing. If it turns to a new direction and it is not facing towards you, you're probably safe and won't need to mill.
Prairie Dragon
Stand away from Dragon and wait until it charges at someone. Begin attacking when it stops its charge. Do NOT windmill it when it is at this spot. When it roars, it is a signal that it is about to charge back to its original location. Windmill it when it is in mid-charge. This will cause it to charge back, though it may spam some fireballs and petrifies before charging back to the party's location. Just wait this out.
This will only work if everyone cooperates and there is no interference.
Giant Lion
Least painful way is to have someone windmill it, fly up on a bird, then drag it away from the other Lions. The other people can then attack without any risk as long as someone on the ground does not windmill. If this method is not possible, then just try your best to not get hit and attack as much as possible.
Mammoth
Spam that mill. Despawns 5 minutes after appearance so try to kill fast.
Giant Sandworm
Windmill it. If it rushes past you during the windmill invincibility, mill again to bring it back. Smash or magnum it twice. Right at the moment it rushes away, windmill then immediately play dead. Cancel play dead when it rushes back and Smash/Magnum again. Do not windmill unless it is rushing away, otherwise the pattern will be thrown off.
This will only work if everyone cooperates and there is no interference.
Yeti
Smash, WM, whatever high damage skill you got. Kill as fast as possible since it despawns 10 minutes after appearance. Finish is based on first hit and damage, unlike the other field bosses.