Make Conjugate[x+y z] into Conjugate[x]+Conjugate[y]Conjugate[z]?

Conjugate[x + y*z] // FunctionExpand
Conjugate[x + y/z] // FunctionExpand

Conjugate[x] + Conjugate[y] Conjugate[z]

Conjugate[x] + Conjugate[y]/Conjugate[z]

Maybe not the neatest but should fit your needs:

foo = # //. (c : Conjugate)[p : (_Plus | _Times)] :> c /@ p &

Conjugate[x + y*z] // foo

Conjugate[x] + Conjugate[y] Conjugate[z]

Conjugate[x + y/z] // foo

Conjugate[x] + Conjugate[y] / Conjugate[z]