I am making a batman-inspired Vengeance Paladin, but I am unsure what her alignment should be. Can someone help me plz?

Probably Neutral Good.  Batman is very law and order focused BUT uses vigilante tactics to enforce that.

Quote from Appellion >>

Probably Neutral Good.  Batman is very law and order focused BUT uses vigilante tactics to enforce that.

 
One does not have to follow the law the be Lawful - the lawful side of the spectrum generally looks at whether you follow some kind of code, which need not be the law of the land.

One of Batman’s key personality traits is his Code - he tends to be unflinching in his views of right and wrong, with a strict set of rules on how he is willing to apprehend perps.  

I also do not think he could be considered Good.  Yes, he does things for the greater good, but he also is strongly motivated by his own selfish desires—it is a common theme throughout Batman that he is partially doing this for his own pleasure, and feels at his best when he is tracking down a sociopath harming the entire city.  He isn’t evil per se, but he is so strongly motivated by self interest and so readily employs aggressive tactics he cannot quite be considered good. 

I would personally ascribe Lawful Neutral to him.-

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All of which, for OP’s purpose, goes to show some of the flows in alignment - Batman is one of the most universally known characters and yet folks can still arrive at different alignment conclusions. 

That is why Alignment works best like any other pseudo-scientific personality indicator - such as Myers-Briggs. You do not want to think of it as an absolute definition of someone’s personality, but a way of short form describing something in (somewhat inaccurate terms).  It can be helpful as a guide to roleplaying, but should not be absolute.

I would not put too much thought into this right now - it should not influence your gameplay much.  Especially for you - alignment is useful when someone does not know what roleplay choice they want to make, so can ask themselves “What would Lawful Evil do?”

You do not need that crutch - you can just ask “What would Batman do?” instead.

True, Alignment is a nice tool, but it’s not perfect.  I dislike Detect Evil & Good only detecting Celestials, Fiends, and Undead (plus Hallowed / Unhallowed Sites), but I can see why they did that.

To me Batman is lawful neutral with good tendency. He followes a moral code and does his best to never sway from it. He is a justicier, a defender of justice that is not affraid to rely on the use of force to overcome villains and even kill them frequently.

Quote from Plaguescarred >>

To me Batman is lawful neutral with good tendency. He followes a moral code and does his best to never sway from it. He is a justicier, a defender of justice that is not affraid to rely on the use of force to overcome villains and even kill them frequently.

 Which Batman are you referring to that's not afraid to kill people? Cause if we're talking about main universe Bruce Wayne Batman, that's literally the one thing he swore he'd never do. 

In Batman cpmics in general he's not afraid to use force to overcome villains, and despite most often trying not to kill villains, he still often did more than a dozen villains he killed include Alfred Stryker Dr Death Karl Helfern,  Mikhail Hugo Stange Mad Monk super srum man is Batman #1 KGBeast Jose Garzonas Dracula, Bleach Thief, Darkseid, Superman The Jocket and a billionaire man In Detective Comics #37 

Neutral Good aligment with Law allegiance, and if we use alternate aligment he would be "material" against esoteric and spiritual because he trust more into science/technology. And here Azrael would be "spiritual" aligment, and Zatanna esoteric.

This is greatly determined by which Batman you are basing him on.

The Animated Series? Lawful Good

Frank Miller's the Dark Knight Returns? Likely CN.

This is so meta. Every alignment thread eventually turns into a discussion of what Batman’s alignment would be, and here’s one for just that. 

To answer the question, imo Batman really highlights the shortcomings of the whole alignment system, as he does not fit neatly into any of the boxes. So my answer is, ask your DM if you even need to write down an alignment. Paladins already have a code of conduct required by their oath, following that is what will be important for the character, not if you write down CG or LN or whatever and then likely don’t ever reference it again. 

Quote from Xalthu >>

This is so meta. Every alignment thread eventually turns into a discussion of what Batman’s alignment would be, and here’s one for just that. 

To answer the question, imo Batman really highlights the shortcomings of the whole alignment system, as he does not fit neatly into any of the boxes.

A huge part of that is because he's had dozens of writers across multiple forms of media for what's approaching a 100 year time frame and he's been reimagined numerous times.

I do think it would be a good idea to consider which iteration of Batman you are wanting to take inspiration from since there are several that do have a variation of different codes, ideals, and personalities. Dark Knight Batman? The Arkham Knights video game Batman? The Batman Animated/Justice League Animated Batman? Batman from a specific era of the Comics. The Character has evolved and changed over time, and between writes that have different visions of the character.

Between the fact that D&D alignment, particularly the law-Chaos axis, is heavily open to interpretation, and that Batman is heavily open to interpretation, you're not going to get a definitive answer.

Consider these things:

• D&D alignment doesn't actually matter for almost anything.
• There's no consequence for choosing the "wrong" alignment, nor is there a consequence for changing it
• Not every moral code even fits into the alignment grid
• Your character is not Batman, you're just taking inspiration from him

Think about your character's moral code. What won't they do even when nobody will ever know? What won't they do in public, but might do if nobody knows? What will they do without thinking, even if it means their death? That sort of thing. You don't need to actually have answers for all the questions you ask. It's a new character; they develop in play. But you do need to have some idea of what they'll do for some of them, or at least what they think they'll do.

Once you've considered these questions, an alignment might stand out to you.

If it doesn't, pick one, and think about it. See if it fits.

If it doesn't, try a few more.

If all else fails, do what everyone else does: Write down Neutral Good, and never think about it again. :)

This is one reason I prefer the Alignment quiz from the Hero Builders Guidebook. Alignment doesn't determine how you act and think. How you act and think determines your alignment

This argument has been raging for as long as Batman and alignment have existed, to the point that I almost think OP is asking just to stir folks up. There is no right answer.

Personally, I don't think characters have alignments. Choices do, and characters are sums of those choices. Plenty of interesting, classic characters are full of inner conflict and thus not all of their actions line up the same. Also, when we read it watch Batman, we filter his choices through our own perspective and thus two people might not ever agree on the exact influence of good/evil/law/chaos on any given decision.

So just pick an alignment that feels right and go with it. Whatever you pick, there's an army of Internet warriors who agree with you.

Quote from scatterbraind >>

This argument has been raging for as long as Batman and alignment have existed, to the point that I almost think OP is asking just to stir folks up.

 I actually was not aware of this argument before, just to clarify

From what I can tell, the basic answer the thread has come to is that no alignment is really right for Batman and it’s better to just get the ideals down (of whichever version of Batman you see as being the “real” one).  If you’re DM is forcing you to choose an Alignment, jot down whatever (probably not Chaotic Evil) and confirm with the DM that you’ll be Role-playing with such and such rules.

Quote from Tanin_Kendrake >>

Alignment doesn't determine how you act and think. How you act and think determines your alignment

 This. All the this. This couldn’t be more correct.

Chaotic good is the only real answer.  He doesn't follow laws at all.  He follows his own moral compass to do what he thinks is right.  

I guess an argument could be made about which batman we're talking about.  The Christopher Nolan Batman is absolutely chaotic good.  The Adam West batman may be lawful good.  I think the general arc of the character as a whole is chaotic good though.

The easy way to handle this is just write in the alignment spot: "BATMAN".  The DM should know relatively speaking what you mean without actually having to describe the system specific alignment.