Sometimes you might pick Eye of Equinox, sometimes you pick sightstone and face. But I don't think ZZot portal is a good pick for supp (unless situational) because it's mostly for top and jungle that don't need extra items like Malphite. I think you should have Locket of Iron Solari (for mr and shield), Frozen Heart (for attack speed) Mikael's Crucible (for cc), and don't forget the boots, mobility (for roaming and warding) or swiftness (to get into fights more quickly). But Banner of Command should be used in the late games, especially when your lanes are pushed in, Randuin's can be used for slowness, and Rylai's scepter is often used for AP support like soraka.
That's pretty much it, if you need help, I'll try my best to answer.
P.S: I play Thresh.