Is this what you wanted?

    expr = E^(-((I β)/2)) p (Cos[α/2] (Cos[θ]^2 + Sin[θ]^2 Sin[ϕ] (-I Cos[ϕ] + Sin[ϕ])) + 
        E^(I β) Sin[α/2] (Cos[θ]^2 + Sin[θ]^2 Sin[ϕ] (I Cos[ϕ] + Sin[ϕ])))

![Mathematica graphics](https://i.sstatic.net/3nscW.png)

    expr /. Complex[x_, y_] :> Complex[x, -y]

![Mathematica graphics](https://i.sstatic.net/T51d9.png)