There are System options one can set to control the behavior of `D`. One of these is the "ExcludedFunctions" option:

	excluded="ExcludedFunctions"/.
        ("DifferentiationOptions"/.SystemOptions["DifferentiationOptions"])

>{Hold,HoldComplete,Less,LessEqual,Greater,GreaterEqual,Inequality,Unequal,Nand,Nor,Xor,Not,Element,Exists,ForAll,Implies,Positive,Negative,NonPositive,NonNegative,Replace,ReplaceAll,ReplaceRepeated}

These are functions that D will not differentiate. We can add Conjugate to this list by using:

	SetSystemOptions["DifferentiationOptions"->
        "ExcludedFunctions"->Union[excluded,{Conjugate}]]

>DifferentiationOptions->{AlwaysThreadGradients->False,DifferentiateHeads->True,DifferentiateIteratorIndexed->True,DirectHighDerivatives->True,DirectHighDerivativeThreshold->10,ExcludedFunctions->{Conjugate,Element,Exists,ForAll,Greater,GreaterEqual,Hold,HoldComplete,Implies,Inequality,Less,LessEqual,Nand,Negative,NonNegative,NonPositive,Nor,Not,Positive,Replace,ReplaceAll,ReplaceRepeated,Unequal,Xor},ExitOnFailure->False,HighDerivativeMaxTerms->1000,SymbolicAutomaticDifferentiation->False}

Now, `D` will not try to differentiate `Conjugate`:

	D[Conjugate[f[x]], x]//InputForm

>Conjugate[Derivative[1][f][x]]

We are now free to give `D` rules for differentiating `Conjugate`:

	Unprotect[Conjugate];
	Conjugate /: D[Conjugate[f_], x__] := Conjugate[D[f, x]]
	Protect[Conjugate];

Let's see what happens to the OP example now:

	D[Conjugate[f[x, y, z]], x]

>Conjugate[(f^(1,0,0))[a,y,z]]