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The code for the problem is:

J = 1    

newcoherentstate[α_, χ_, 
 m_] := ((ψ[α, χ])/(1 + 
  Abs[ψ[α, χ]]^2))^(J) *(ψ[α, \
   χ])^(m)*Sqrt[Binomial[2 J, J + m]] ;

NewMatrix [α_, χ_] := 
 Table[newcoherentstate[α, χ, m], {m, -J, J}];

NewMatrix1[α_, χ_] := 
 ArrayReshape[NewMatrix[α, χ], {1, 2*J + 1}];

NewMatrixC[α_, χ_] := 
 Conjugate[NewMatrix1[α, χ]];

Using matrix form, the result is:

enter image description here

What is making the Conjugate appear everywhere? I simply want the Conjugate function to take the complex conjugate of each term in the row vector.

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What you observe is the fact that Mathematica assumes the most general case, since you did not specify any assumptions. If you know that e.g. \[alpha] is real-valued, you need to let Mathematica know this very important fact.

See this:

z = Exp[I*a];
Conjugate[z]

E^(-I Conjugate[a])

This is basically what you asked about in your question. IF a is real, then you need to provide this as Assumptions to e.g. Refine or Simplify (depending on your expression, Simplify might take more time than Refine due to its nature):

Simplify[Conjugate[z], Assumptions -> Element[a, Reals]]
Refine[Conjugate[z], Assumptions -> Element[a, Reals]]

both yield

E^(-I [Alpha])

as you would expect, knowing that a is real. If this assumption is true for all your calculation you might want to globally specify this assumption by executing

$Assumptions=Element[a,Reals]

before you start over with your symbolic calculus.

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