Find the First Bracket Match

Brain-Flak, 685, 155, 151, 137 bytes

(())({<{}({}()<(()()()())>)({}(<>))<>{(({})){({}[()])<>}{}}{}<>
([{}()]{})(({})){{}({}[()])(<()>)}{}(<>)<>{{}<>{}({}<>)}{}(<>[]<>)>()}<>)

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136 bytes of code, plus one byte for -a. One indexed.

530 bytes golfed off! That's probably the largest golf I've ever done.

14 bytes saved thanks to Riley!

This abuses a formula of the opening/closing parenthesis: if you take the ASCII values, increment it by one, and take modulo of 4, the openers (({[<) will always get 0 or 1, whereas the closers ()}]>) will always get 2 or 3.

Explanation:

#Push 1
(())

#While true
({<

    #Pop stack height
    {}

    #Compute (TOS + 1) % 4
    ({}()<(()()()())>)({}(<>))<>{(({})){({}[()])<>}{}}{}<>([{}()]{})

    #Decrement if positive
    (({})){{}({}[()])(<()>)}{}

    #Push 0 onto alternate
    (<>)

    #Toggle back
    <>

    #Pop two zeros from alternate if closer
    {{}<>{}({}<>)}{}

    #Push height of alternate stack
    (<>[]<>)

#Make each time through evaluate to 1
>()

#Endwhile
}

#Push the number of loops onto the offstack
<>)

05AB1E, 17 16 10 bytes

-1 thanks to carusocomputing

-6 thanks to Adnan for his amazing insight that "after incrementing, the second last bit is 0 for an opening bracket and 1 for an closing bracket"

Ç>2&<.pO0k

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Ç          # Get input as ASCII values
 >         # Increment
  2&       # And with 2 (0 for open and 2 for close brackets)
    <      # decrement 
     .p    # prefixes
       O   # Sum
        0k # Print the index of the first 0

Vim, 23 bytes

:se mps+=<:>
%DVr<C-a>C1<esc>@"

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I'm really sad about this answer. This solution is beautifully elegant and short, but, by default, vim does not consider < and > to be matched, so I need 13 bytes of boilerplate code. Otherwise, this would just be 10 bytes.

I would have posted a V answer, but that would only be one byte shorter, namely changing Vr to Ò, since Vr is a common vim-idiom.

This is 1-indexed but could be trivially modified to be 0-indexed by changing the 1 to a 0.

:se mps+=<:>        " Stupid boilerplate that tells vim to consider `<` and `>` matched
%                   " Jump to the bracket that matches the bracket under the cursor
 D                  " Delete everything from here to the end of the line
  V                 " Visually select this whole line
   r<C-a>           " Replace each character in this selection with `<C-a>`
                    " This conveniently places the cursor on the first char also
         C          " Delete this whole line into register '"', and enter insert mode
          1<esc>    " Enter a '1' and escape to normal mode
                @"  " Run the text in register '"' as if typed. Since the `<C-a>` command
                    " Will increment the number currently under the cursor