Python, 51 bytes, cracks M Virts

lambda a,b:getattr(__import__('math'),'atan2')(b,a)

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J, 11 bytes, cracks Bubbler

_&,^:_1.&*. or _&,^:_1&.*. or _.&,^:_1&*.

The only thing that could be relevant to complex angle in the puzzle is *., that decomposes complex number into length and angle, then the presence of ^ and : implies it is likely ^:_1 has been used, and associate that with the presence of , and _ which the later can only be used to indicate a number "plus infinity", it would be apparent that _&,^:_1 is used to take the second result returned by monadic *..

However, after formulated _&,^:_1&*., there is still one . left, interesting enough, this . can be combined with _ or _1, both accepted as valid literals by J (_. is NaN, _1. is also negative one but in float format) or forms under &., that utilizes the fact the inverse of *. on a real number is the identity function, results in three different valid answers.

Wolfram Language (Mathematica), 48 bytes, cracks att

N[ToPolarCoordinates[List[##,##]][[Abs[2]2]]]&

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All credit goes to Greg Martin for actually finding the initial solution.

Go, 69 bytes, cracks @bigyihsuan

import."math/cmplx";func Posswf(n complex128)float64{return Phase(n)}

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C (gcc), 68 bytes, cracks? Blue

#include <complex.h>
float h(float n, float at){return carg(n+at*I);}

-- Am I right to believe that for imaginary reasons, a space character transformed into a semi-colon?

Python, 50 bytes, cracks @CursorCoercer

lambda e,f:log(e+(1.0j*f)).imag
from cmath import*

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C (gcc), 65 bytes, cracks Blue

#include <math.h>,x
float h(float y, float s){return atan2(y,s);}

the elevation of ,x was quite a surprise

Wolfram Language (Mathematica), 31 bytes, cracks CrSb0001

NumberForm[Im[Log[N[x+I*y]]],6]

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Wolfram Language (Mathematica), 53 bytes, cracks Greg Martin

Im@Integrate[1/e,{e,1.,#+#2I+Boole[#<0&&#2==0]/9!I}]&

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I couldn't figure out what the 9 was doing there until I reread the question...

Wolfram Language (Mathematica), 57 bytes, cracks CrSb0001

f[x_,y_]:=N[Arg[x+y*I]+(b-b)/6*FIILNR[e,e][][]*mmoorsuxy]

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Be careful what you allow :)

Uiua 0.10.0, 14 bytes, cracks @Tbw

⍥¬0(÷×.±↥⊃∩⌵∠)

Try on Uiua Pad!

Wolfram Language (Mathematica), 13 bytes, cracks Greg Martin

Re@*N@*ArcTan

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Prints more than six significant figures on TIO, possibly because of the Print function or just... I dunno. :P

JavaScript (Node.js), 51 bytes, cracks @l4m2

eval(unescape`%4d%61%74%68["a%74a%6e"+7%5%5%8%30]`)

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