Pudding farming may be difficult to set up, but it does work
It is difficult to actually gain the split ability, but there are a few ways to do this. Be aware that polymorph is insufficient, as
It also gains all extraordinary special attacks possessed by the form but does not gain the extraordinary special qualities possessed by the new form or any supernatural or spell-like abilities.
Again, there are other ways to do this, though, and Master of Many forms is one of them (at 8th level, when you can wild shape into oozes, or 7th level if you can ooze shape earlier for some reason).
If a character does gain the split ability, then they have the split ability. There's nothing that really seems to indicate that the split off clone is in any way a different kind of creature from the original-- a transformed mage who splits becomes two identical transformed split mages with half the hp.
The split off creature may or may not be affected by any ongoing spells, like Shapechange, affecting the 'original' creature, depending on what portion of the self you consider important for the purposes of two creatures being identical, and whether the split results in two new creatures or just one. Any spells not applied would also see their effects not-applied, so if you transformed to gain the split ability via a spell and the copy did not get a copy of that spell, it would also not be transformed. If a copy of a spell is gained, it ends when that spell would normally end.
The split off creature shares most of your equipment. Most methods of transforming are subject to the following:
Creatures who polymorph keep their worn or held equipment if the new form is capable of wearing or holding it. Otherwise, it melds with the new form and ceases to function for the duration of the polymorph.
Since such equipment explicitly becomes a part of the form for the duration of the effect, the duplication via split will also duplicate all such equipment. Such equipment may not be easily accessible to either creature, however, as when equipment unmelds is unclear within the rules. Generally, melded equipment ceases to be melded when a creature ends the effect that caused the equipment to be melded in the first place, but the rules don't actually require that so a GM might prohibit both creatures from ever being able to access their melded items that way, for whatever reason. Anything that causes a creature to revert to its 'true form', however, will unmeld the equipment, and the alter self spell provides explicit unmelding circumstances as well.
The split off creature obviously shares any templates it parent had, since the creature would not otherwise be identical to the original.
The split-off creature has no intrinsic loyalty to nor animosity towards its parent, and shares all of the parent's memories, personality, and alignment, though it possesses its own soul.
You might die, though
It is possible that the split results in two children and the death of the parent-- that is, the spell results in the creation of two new souls and the original creatures soul, lacking a place to go, is slain. Should a creature be slain in such a way, it could be easily raised by spells such as resurrection, though such abilities require that at least one of the children be slain so that their parent can be raised. Only True Ressurection allows the parent, in this case, to return to life without the death of one of the children. An appropriately prepared mage, though, could instead make use of the clone spell. This interpretation, with two children and a dead parent, relies upon the following text:
Instead the creature splits into two identical puddings, each with half of the original’s current hit points (round down)
Which implies that neither of the two created puddings is the original, as well the line from magic jar stating that "Any life force with nowhere to go is treated as slain." The application of this last line is quite shaky, and the DM is free to process the bodilessness of the original creature in any number of alternative ways.