Can you use a fly speed underwater? How does the spell Overland Flight interact with underwater movement?
[RPG] Overland Flight underwater
flightpathfinder-1espellsunderwater
Related Solutions
Firstly, you didn't ask, but movement is effectively cut in half, rather than quartered. Per the rules on climbing, swimming, and crawling:
Each foot of movement costs 1 extra foot (2 extra feet in difficult terrain) when you’re climbing, swimming, or crawling. You ignore this extra cost if you have a climbing speed and use it to climb or a swimming speed and use it to swim.
So movement is ⅓ if the heroes are in difficult terrain - normal swimming is only ½.
Next, spellcasting. There are no specific rules about underwater spellcasting, but the description of verbal components says:
Most spells require the chanting of mystic words. The words themselves aren't the source of the spell's power; rather, the particular combination of sounds, with specific pitch and resonance, sets the threads of magic in motion. Thus, a character who is gagged or in an area of silence, such as one created by the silence spell, can't cast a spell with a verbal component.
It's pretty reasonable to extend this to a character underwater, so spells with verbal components are out. There doesn't seem to be anything else that would restrict underwater spellcasting, however.
Finally, breathing mechanics. The rules for suffocating are on page 183 of the PHB, and are much simpler:
A creature can hold its breath for a number of minutes equal to 1 + its Constitution modifier (minimum of 30 seconds).
When a creature runs out of breath or is choking, it can survive for a number of rounds equal to its Constitution modifier (minimum of 1 round). At the start of its next turn, it drops to 0 hit points and is dying, and it can't regain hit points or be stabilized until it can breathe again.
For example, a creature with a Constitution of 14 can hold its breath for 3 minutes. If it starts suffocating, it has 2 rounds to reach air before it drops to 0 hit points.
The GameMastery Guide on Water says, "The rules presented in the Pathfinder RPG Core Rulebook for underwater combat apply to creatures not native to this dangerous environment [i.e. underwater], such as most PCs" (214). (Fortunately, monsters can be sorted by terrain.)
It seems reasonable that an eidolon that possesses gills be considered native to water, and, therefore, such an eidolon shouldn't suffer penalties when using its natural weapons during underwater combat, but you should check with your GM to be sure.
Also Paizo creative director James Jacobs in a 2012 post said, "Creatures that have the aquatic [sub]type don't have to worry about those [underwater combat] penalties[,] but all manufactured weapons must abide by these rules, regardless of whether or not the creature wielding them is aquatic." So, in conjunction with the GameMastery Guide's rule, if, for example, the eidolon opts for a greatsword instead its claws, the eidolon suffers the underwater combat penalties on attacks made with that weapon. (Note that Jacobs makes it clear that this is a ruling not a rule, but it seems okay and comes from an often authoritative source.)
Best Answer
Fly speed won't help you at all while underwater.
Unless you have some kind of ability that allows your to use your fly speed while swimming, your different speed types do not help you on different terrains. Swim speed will not help you move on land or air, land speed won't help you move on water or air, and thus, fly speed won't make you run faster on land, nor make you swim faster on water.
Excluding the swim skill rules, which allow you to use half your base land speed as your movement rate, but does not grant you swim speed at all. But that's a specific rule for the skill.
To clarify, whenever a rule is talking about your speed or your base speed, it is a reference to a character's base land speed, not whichever speed is highest. This can be found on common game terms:
Here is the basic description of the three movement types:
If we check the Fly skill description, it says:
Finally, the Swim skill makes several mentions of your speed (and even your base land speed on the unchained rules), but nowhere it talks about using other speed types while underwater.
There are exceptions, of course. The Swim on Air spell allows a water-based creature to use her swim speed to fly as if swimming on air, for instance.
However, it isn't unheard of that a creature that could make more sense to use a certain speed type instead of another. Some water-based creatures have zero land speed, so the GM might have to give it at least 5 land speed when trying to do something extraordinary while on land, even though normally they cannot move on land at all. Or even house-rue to use a speed type as another if that would make more sense in a given situation.
Example, the Octopus has a special water-based movement ability called Jet, which allows them to move 200 ft in a direction, while this ability does not mention that it can be used on land (even if that doesn't make a lot of sense) it's perfectly fine to allow a flying octopus to use that ability while airborne.