I figured it out. No, the doom doesn't get all the zoanthrope powers.
See, warp field, on pg 44 of tyranid codex, says "zoanthropes get a 3++ save"
Likewise, quite a few of the Hivetyrant powers on Pg 62, say the "hive Tyrant blah blah..."
So a WAAC rules lawyer type could argue that the warpfield doesn't do anything for the Doom, and likewise the Hive Tyrant powers do nothing when used by the swarmlord, even though that's stupid. I imgaine there was a similar problem with old one eye and Living Battering Ram.
Q: Tyranid Special Characters. Are Tyranid Special Characters
considered to be special versions of the standard creatures of their type
for the purposes of using psychic powers, biomorphs and special rules?
For example, is the Swarmlord considered to be a Hive Tyrant for the
purposes of using Hive Tyrant psychic powers? Similarly, is
Deathleaper considered to be a Lictor for the purposes of Chameleonic
Skin and Pheromone Trail, is the Doom of Malan’tai considered to be a
Zoanthrope for the purposes of Warp Field, is Old One Eye considered
to be Carnifex for the purposes of Living Battering Ram and are
Ymgarl Genestealers considered to be Genestealers for the purposes of
Brood Telepathy? (p56-61)
A. Yes in all cases.
So all they're saying is that yes, all these special variants can actually use the special rules they are given, even though they reference the base type. Brrod telepathy references genestealers, not "ymgarl Genestealers" and some jerk was trying to argue that they don't get the benefit, then. That's why it says "use". It still counts when used on or by them.
It does NOT give extra powers to the Doom or anyone else. The unit entry lists quite clearly what psychic powers the doom gets (exactly one, btw) and swarmlord does as well (four).