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Imagine a four armed woman, who stands approximately 10ft tall. For whatever reason, this woman lives a life where fights are reasonably common and she decides to start taking some self defence classes.
EDIT: She's not unique, there are people like her who could come up.
The problem
IMO the reason we don't see more two weapon fighters is because our brains are not sufficiently parallelized for calculating the motions of two objects at a time. Putting this in simpler terms, adding the second weapon decreases our effectiveness with the first.
Diminishing returns
I think increasing the number of arms to 4 suffers from the law of diminishing returns. In most cases, without special training, just using 3 shields and blocking as many directions as possible, while your primary hand wields your weapon will be the best fighting technique.
Real fencing
I've actually fenced competitively. Once I start a lunge, I've already decided all of my measures and counter-measures. I go through them as fast as I can without awaiting response from my opponent. I use the time in-between attack and perry to analyze what I and my opponent have been doing and are likely to do and try to figure out my next attack and defense combinations prior to the next engagements. I do not know whether high end fencers do this the same way or they are able to react to the actions of the other fencer in real time.
A master
Someone who spent years trying to master multi-armed combat might be able to do better, perhaps much better. Even if she couldn't, if she could master a series of automatic attacks and defenses with 3 arms with which she'd simply use muscle memory to control it would provide her with a big advantage.
Meanwhile she could concentrate on the attack and defense of her main weapon.
Fighting someone like this
As someone who would have to face her, you would need to worry about what all of those "automatic" arms are doing because they can still harm or kill you. Even if they're not a huge threat (compared to her primary weapon), they'll still be a huge distraction which probably gives her primary attack (her main hand) a huge advantage.
Some fighting styles
Ranged:
Just remember you'd have difficulty aiming more than one of these at a time. Although especially for the primitive ranged weapons, she could spend all of her thinking doing the aiming and then using her arms to bring the next weapon into the ready position - almost doubling her rate of fire.
Melee:
Note that there's always a (main) weapon. This is the one she spends the most time thinking about and using to react to her opponent. The others she'll use reflexively to block blows or occasionally take advantage of an opening.
I think these last two combinations could be particularly deadly. You hold back on the spear, fighting mainly sword + shield. When you're opponents backs up for a short breather and provides an opening, you have the range to suddenly jab and attack that opening.
It would probably quite deadly to those not used to the technique.
An analogy
I fenced right-handed. My roommate fenced left-handed (because he was left-handed).
He loved fencing right-handed people because they were used to fighting right-handed people and they were unfamiliar with left-handers. He was used to fighting right-handed people and almost always won in such matches.
I hated fighting left-handed people for all the same reasons.
In your Universe, it might be the same for your protagonist. For people unfamiliar with her style of fighting, she'd be extremely deadly. Meanwhile, if she hasn't been around any of her own kind for a while, she might suffer problems when she's faced with fighting others like herself.
Fighting a guy with four arms involves a lot of getting hit in the face with two fists at one time, and getting held off of the ground by one set of arms while taking upper rights and lefts to the head without being able to block.
If you want to get an idea of what fighting with 4 arms looks like, just watch some videos of Goro fighting. Game developers have spent many years developing a four armed fighting style.
As far as what styles she would be naturally be good at: boxing and judo. Throwing any combination of 4 fists in someones face would be hard to counter, and an extra set of arms would open up a whole new range of grappling techniques that a judo master could only dream of. Fighting two armed species would be like fighting someone with no arms at all, two arms would hold onto their two arms while the other two did the pummelling. The biggest advantage of fighting with four arms would be the ability to attack and defend at the same time (just cover your face and go in swinging with your other set of arms).
There are lot's of other examples of 4 armed fighting already in pop culture, including General Grievous from Star Wars:
The 4 armed Green Martians from John Carter (who also happen to be 10ft tall):
And General Krell from Clone Wars:
Watch one of the most intense four-armed fight scenes ever:
https://youtu.be/oPvNkIPFdI8?t=45s
I agree that there are no specific styles that fit that woman, but disagree with other people reasons.
Efficient fighting styles (including judo, karate, fencing, etc.) emphatize the need to use the whole body to impulse the attack. For example, you could try to throw a punch:
Keeping your torso static and just moving the arm.
Keeping your feet static (but a little separated) and rotating the hip while launching the arm.
Advancing the foot (of the same side that the arm throwing the punch) at the same time.
Without any training, is should be easy to yourself than the first option is way less powerful, and that the last one is the best.
Of course, I am not saying that the four arms (if can be properly coordinated) are not an advantage; in a martial art you could use two of them to prevent the opponent form using his own arms to block your act, in judo 1 you could get a firmer grip on the rival's judogi, in fencing.....
Additionally, some martials arts (like Taekwondo) would nullify that advantage, as they rely mostly in kicks.
1As a former judo practicioner, Judo is more of a sport that incorporates some martial arts elements that a martial art itself, because:
It is difficult to use judo against someone if the other does not use judo (there are no counters against illegal attacks, as kicks or punchs)
It is entirely one against one, if you get in a "two against one" fight, you are completely defenseless against the opponent you are not gripping
Even if the opponent makes a move that lets you throw him away, it is extremely dangerous throw an untrained person against a hard surface (it is dangerous even if the person is trained). Unless it is a live or die situation, you should never do that.
From the beginning, it was designed to be an sport.
1) There are no existing fighting styles for people with two sets of arms because such people do not currently exist. That being said, if such a group of people were to exist and mirror human history and societies, there would probably be a great number of fighting styles that she could chose from. With regards to the concept of being 'naturally' good at a certain fighting style or method over another, the concept is a fallacy - one cannot be naturally good at such a thing, all styles require hard work, training, and practice, whether they are hard or soft styles. Some people do have naturally greater control over their body mechanics, but this would lead to general base line advantage in all fighting styles, not one or another. Depending on the opponents that she is facing however, certain styles would provide her with an advantage over others. If she is facing those generally weaker and smaller than herself, hard styles like karate and tae kwan do would be advantageous, where as if she is smaller and having less strength, fluid and soft styles would be advantageous in that fight by using an opponents energy against them. In the end, the choice of fighting style is more likely going to be based on personality, desired effect, and expected opponents rather than her body having an extra set of limbs over a human.
2) Working with the assumption that she is not completely unique and that there are other four armed persons that have designed fighting styles previously, there is likely to be a wide range to chose from. However some common elements that would most likely exist given an extra set of limbs would be the ability to both defend and attack at the same time, the management of limbs to prevent tangling, misdirection (using one limb to obfuscate the actions of another), and using two or more limbs in conjunctions to strike multiple locations at once. I would also expect that given the natural number of weapons provided by an extra set of arms and the balance issue their additional weight above the waist, there would be a reduced focus on kicking, jumping, and even movement in general, most likely preferring low stances and focus on grounding.
Martial arts are all about body mechanics and using your body to make an effective offense or defense, minimizing injury to yourself with maximizing disruption to your opponents. If you really want to develop an intricate martial arts style for your character or group of people, you will need to first resolve issues of anatomy, as that will greatly effect how they fight. Such things as how does the shoulder on the lower arm function, whats it's range of motion? Is the upper arm effected by the presence of the lower arm? Is their torso longer and have different skeletal and musculature structure than a humans? All of these things will effect their final fighting style. Martial arts were generally developed by groups of people being denied access to military arms and armor in order to resist against those that sought to do them harm, so there might even be a need to understand the history of the Martial Art in order to understand it's techniques. Was it created during a period when peasants were protecting themselves against armed bandits, or was it created primarily as a method of meditation on the natural world and adapted to the purpose of self defense?
Jim2B offered a great explanation about the problems when using weapons. Even though, martial arts do not pose that problem.
For people who had practiced some martial arts, since left and right hand have a different use depending on your guard, you can take full advantage of both arms.
The same would be true with four arms. So the key is to give different tasks to each arm. One great example would be creating an alternative of boxing.
Still, though, it must be taken into account how tall this heroin is: 10ft tall makes the fighting against standard humans really different.
So, implications of 4 arms:
Implications of 10ft:
How I would combine both?
