Do alchemist bombs do extra damage on a confirmed critical hit?
Is there any way an alchemist/rogue character can get sneak attack damage with bombs?
As Red Metal said, alchemist bombs do extra damage on a critical hit, but it's not noteworthy or particularly useful.
Bomb (Su): In addition to magical extracts, alchemists are adept at swiftly mixing various volatile chemicals and infusing them with their magical reserves to create powerful bombs that they can hurl at their enemies. An alchemist can use a number of bombs each day equal to his class level + his Intelligence modifier. Bombs are unstable, and if not used in the round they are created, they degrade and become inert—their method of creation prevents large volumes of explosive material from being created and stored. In order to create a bomb, the alchemist must use a small vial containing an ounce of liquid catalyst—the alchemist can create this liquid catalyst from small amounts of chemicals from an alchemy lab, and these supplies can be readily refilled in the same manner as a spellcaster's component pouch. Most alchemists create a number of catalyst vials at the start of the day equal to the total number of bombs they can create in that day—once created, a catalyst vial remains usable by the alchemist for years.
Drawing the components of, creating, and throwing a bomb requires a standard action that provokes an attack of opportunity. Thrown bombs have a range of 20 feet and use the Throw Splash Weapon special attack. Bombs are considered weapons and can be selected using feats such as Point-Blank Shot and Weapon Focus. On a direct hit, an alchemist's bomb inflicts 1d6 points of fire damage + additional damage equal to the alchemist's Intelligence modifier. The damage of an alchemist's bomb increases by 1d6 points at every odd-numbered alchemist level (this bonus damage is not multiplied on a critical hit or by using feats such as Vital Strike). Splash damage from an alchemist bomb is always equal to the bomb's minimum damage (so if the bomb would deal 2d6+4 points of fire damage on a direct hit, its splash damage would be 6 points of fire damage). Those caught in the splash damage can attempt a Reflex save for half damage. The DC of this save is equal to 10 + 1/2 the alchemist's level + the alchemist's Intelligence modifier.
Alchemists can learn new types of bombs as discoveries (see the Discovery ability) as they level up. An alchemist's bomb, like an extract, becomes inert if used or carried by anyone else.
Splash weapons (such as bombs) cannot normally be used with sneak attack. You would need a special ability to say you could sneak attack with splash weapons, like the mentioned Underground Chemist. I don't know if any other options exist to do it. For what it's worth, it's not a good idea to build for. Ranged sneak attacks are incredibly hard to pull off, as you must be within 30ft and cannot use flanking to qualify for sneak attack.
So a critical hit with a bomb gets an extra 1d6 on the primary target only?
yup
So a critical hit with a bomb gets an extra 1d6 on the primary target only?
And an additional INT modifier of damage, since that is also part of the attack's base damage.
I don't think you can use Vital Strike at all, because throwing bombs is either a special standard action or a full-round action. You never use the standard attack action that Vital Strike requires.
So a critical hit with a bomb gets an extra 1d6 on the primary target only?
Plus various damage bonuses - INT from Throw Anything, the +1 from Point-Blank Shot, a Bard's Inspire Courage etc. - all get multiplied as well.
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Yeah, the static bonuses can get up there if the character is built right and has a good party makeup.
So, you are at a level that give bombs 2d6 damage, and have a static of +3 on top. The target that is critted is damaged for 3d6 +6 for the critical and the splash damage is the minimum damage for the bomb (5) with a reflex for half (2).
Ranged sneak attacks are incredibly hard to pull off, as you must be within 30ft and cannot use flanking to qualify for sneak attack.
Since the inclusion of Expert Sniper, it's actually not that difficult to pull off, and since Bomber gives you bombs that deal damage equal to your sneak attack dice, you can get some fairly decent damage from a single attack.
Is it possible to take Improved Critical (Bomb)?
Improved Critical requires you to select a weapon you are proficient with.
Alchemists are proficient with bombs, and ithey count as a weapon. So yes, you totally can!
It's probably not worth it (only 20/x2 and most of the damage is not multiplied), but definitely possible.