Beidou (北斗) is captain of the Crux fleet Alcor ship.
Like Yun Jin's skill, this is a parry. At Lv13, the scaling on a perfect parry (or when max of 2 hits taken) is $$258.4+680 = 938.4$$ which is quite high for a 7.5s cd (it is a nuke). Basically a perfect parry will quadruple the damage compared to not parrying at all. The skill also has greater AoE on a perfect parry.
Assuming 4p emblem with 46.6% dmg bonus (+24% dmg bonus from ascension stat), the dps of a perfect parry is $$x_e = (1.466+.24)*938.4/7.5 = 213.45$$
Somehow, Beidou, Qing, and Cyno all are electro and have 7.5s cd on their E. As for E scaling, Beidou > Qing > Cyno (outside burst), which looks like a cruel joke to Cyno. But when Cyno uses his burst, his E becomes buffed immensely (500+ scaling nuke every 4s).
Her q will bounce between multiple enemies. Her C2 increases the number of discharges. At level 13, the scaling on a lightning discharge is \(d_0 = 204\)%. There is at most one discharge per second, and discharges proc when normal and charged attacks hit (it can proc on Tartaglia's E attacks but not Raiden's Q attacks; this is Genshin's way of balancing Raiden).
Each time it procs it will jump to up to 3 targets (which can be the same) per second. With her C2, it can jump to up to 5 targets per second. This means the total damage per enemy is bounded above by a constant. The burst lasts 15 seconds with an 80 energy cost and 20s cd.
Suppose Beidou is Lv. 90 (has 24% Electro DMG bonus from ascending) with 4 piece Emblem with Electro DMG Bonus goblet and 200% ER. The total DMG Bonus on the burst is \( 200(.25) + 46.6 + 24 = 120.6 \) The MV on a discharge is then $$d = 2.206d_0 = 450.024$$ and this can occur at most once every second. In practice it is possible to proc the discharge around 10-12 times. With fast attackers like Yoimiya etc. it is possible to proc it more. Slower attackers like claymores will proc it less. So against a single target, if it procs around 11 times on average, for 15s with a 20s cd, the DPS is $$x_1 = 11d/20 = 247.5132$$ Beidou excels more in AoE situations, when there is more than one enemy. Note the calculation does not factor the initial hit of the burst (scaling 258.4% which averages to around +13 DPS).
For 1 enemy, scaling is d. For 2 enemies, scaling per enemy is 5d/2 as now the discharge can start bouncing between enemies. For \(n\ge 2\) enemies, the scaling per enemy is \(5d/n\).
So Beidou's DPS per enemy peaks when there are exactly 2 enemies. This situation happens in various domains and iterations of the Spiral Abyss. In this case the DPS becomes $$x_2 = 5x_1/2 = 618.783$$ So Beidou has high DPS without reactions, and this is an off-field ability.
Note for more than 2 enemies the damage per enemy decreases (scaling is sub-linear). For example, for 3 enemies the DPS is $$x_3 = 5x_1/3 = 412.522$$
It may be shocking how high Beidou's dps is for a 4 star. Even without reactions, she beats many 5 star characters including Hu Tao and Ganyu. As an Electro DPS Beidou is clearly better than Keqing, and in AoE situations can even be a better DPS than the Raiden Shogun. Her burst is simply broken.
Note that Beidou's Q has normal ICD (follows a 3 hit rule), so due to how frequently the discharges attack in a given second, usually she will be the one triggering Electro reactions, meaning her EM is factored into the reaction, unless she is paired with a character that can attack faster than or keep up with Beidou (a C6 Yoimiya might). However, since Beidou does so much damage herself, usually one wants to build more CRIT, ATK, DMG Bonus than EM for Beidou.
The first hit of Beidou's q applies 4C electro (when she slams her sword into the ground). The discharges apply 1A electro. Her q snapshots and also provides DMG reduction. With her C1 her q generates a shield that scales with her HP.
For how much damage and utility it provides, her Q Stormbreaker qualifies as one of the best bursts in game and solidifies her position as one of the best sub-DPS characters.
Beidou's Q is an example of a bounce effect (similar to Asta's skill in Star Rail). Electro-charged and baizhu's E are other bounce mechanics. Even though, bounce damage can hit multiple opponents, it basically has the same scaling as single-target damage (sublinear), whereas most aoe abilities have genuine linear scaling. But bounce effects are still better in multi-target situations than single-target damage as bounce effects can at least hit multiple enemies.
Nahida's E is similar to a bounce effect in that it can hit multiple enemies but has sublinear scaling due to the upper limit of marking 8 enemies. (In practice this does not even matter since most aoe situations in game will not have more than 8 enemies concurrently.) For more info, see Scaling. The aoe scaling hierarchy is: single-target < sublinear (bounce) < linear (most aoe) < quadratic (Childe).
4 piece Emblem (released in Version 2.0) is her bis. 4p Gilded can be used for quicken builds.
For most builds, use ATK sands, Electro DMG goblet, CRIT circlet. Substats: CRIT > ATK, ER. In some hyperbloom setups, full EM may be viable.
5 star weapons: Wolf, Skyward Pride, Redhorn, Beacon. 4 star weapons: Sea lord, Serpent Spine, Favonius.
While Beidou could act as a hyperbloom driver (full EM build, mailed flower), it is almost a waste of her talent. Her Q has high bounce scaling as calculated above, so it is almost always better to build atk-electro-crit to maximize that raw damage. Moreover, her Q costs energy, whereas Raiden E, Yae E, Kuki E do not and can all be used to drive hyperbloom instead.
Main teams archetypes include quicken, Eula, fireworks, taser. For electro batteries, Yae, Raiden, Fischl, Kuki can work. Note Beidou Q can proc on both NA and CA.
Some example teams include: Cyno-Beidou-Nahida, Itto-Beidou, Childe-Beidou fireworks, Eula-Beidou-Raiden, Scaramouche/Heizou/Sucrose-Beidou-Yelan-Nahida/Fischl taser, Beidou-Nahida-Yelan/XQ-Bennett rainbow national, Beidou-Yaoyao-Sucrose-Yae/Fischl (spread taser), Dehya-Nahida-Childe-Beidou (rainbow Childe), Alhaitham-Beidou.
Beidou tends to work very well with Alhaitham.
The following 4 star quick-taser team has been able to clear abyss versions: Beidou-Yaoyao-Sucrose-Fischl. Here Sucrose acts as a driver while Beidou and Fischl provide off-field electro sub-dps, and Yaoyao provides healing and dendro app. Sucrose can use sac frag and full EM build, and her swirls can proc aggravate too.
An upgrade of this team might replace Fischl with Yae. Nahida and Scaramouche can also be used. For example: Beidou-Nahida/Yaoyao-Scaramouche/Sucrose/Jean-Yae/Kuki. Perhaps a c6 kazuha can be used as an anemo driver too for the Alcor duo to unite in a team. But in that case, even Xiao can act as one.
One can try to use Raiden in place of Beidou, but her E may not provide as much sub-dps as Yae or Beidou (as we are not discussing hyperbloom teams here). Raiden's dps comes from her burst, and if an anemo E sub-dps releases, one could build teams like: Raiden-Nahida-Anemo E subdps-Yae/Kuki. Currently teams such as Raiden-Nahida-Venti/Sucrose/kazuha-Yae/Kuki can be used. However, the off-field anemo app comes from the bursts of Venti/Sucrose/kazuha. It costs energy, whereas something like Nahida, Yae, or Dehya E does not cost energy. The release of an anemo subdps could add more convenience to teams such as this, especially if they have long duration and 100% uptime on E like Raiden, Nahida, Yae. Such a unit could be great for Cyno too (persistent vv shred) if their E lasts 20s or more.
The Dou in Beidou's name could be a reference to a certain empress during the Tang dynasty.