IMPORTAAAANT NOTEEEE!!
Advent Children is a movie that doesn't have a determined storyline. You have to read between the lines, and just as Tetsuya Nomura said, it's a movie that doesn't explain everything - thus making it possible for the audience to make their own conclusions. Parts of the information below are MY assumptions, so don't flame me if you don't agree. On the other hand, if you can PROVE that I've gotten something wrong, you can flame me =P
We learned early that Kadaj is the character who "holds the key to the story", and very much so.
Kadaj is not a real person. And I don't mean like duh-he-is-a-character-in-an-animated-movie kind of unreal person =P
He is one out of three entities of Sephiroth, the other two being Yazoo and Loz. This is possible because after Sephiroth was defeated, he refused to enter the Lifestream. Although he was divided into different parts, his will and spirit remained inside these new creations of contaminated Spirit Energy.
Being the spirit of Sephiroth, Kadaj is constantly struggling with his own will, as well as Sephiroth's. Even though Kadaj has developed personality and qualities, he is still a puppet controlled by Sephiroth.
In contrary to common belief, Kadaj and his "brothers" do not have Jenova cells inside them. The thing that proves this the best is the fact that Yazoo actually ASKS the Turks if "Mother" is inside/on top of the Meteor Memorial statue. If he really did have Jenova cells, he would sense "her". The reason they are there in the first place, is because the children led them there. After stepping into the lake in the City of the Ancients, Kadaj tainted it with his will. The children drank from it, and thereby activated the Jenova cells inside them. The SHM did this in purpose of waking the Reunion instinct within the children, and thus they would be led to Jenova.
Kadaj's constant search for his "Mother" are caused by two things.
1. His own yearn for a mother who can take care of him and give him the love he has never felt, but so badly desires. Keep in mind, that Kadaj is a very young boy (2 years old, to be precise), and children needs a kind of affection that only their mother can give them.
2. Sephiroth's desire to find Jenova, so that when Kadaj merges with them, Sephiroth can return. So, he is looking for Jenova because he is simply carrying out Sephiroth's will.
Kadaj knows he is remnant of Sephiroth. Although he never knew him, he can still feel him, and he doesn't like that. He is afraid that "Mother" loves Sephiroth more, and that it will make him able to return. Knowing this, you understand what kind of confused state Kadaj must be in. One part of him wants to find mother, while another doesn't want Sephiroth to return. Both reasons are thoughts made up by a child - no child wants their mother to play favourites.
When Kadaj finally obtains Jenova's box, his journey is over. He's done what he set out to do - he has found his mother. With that being done, he no longer knows what to do. When he talks to Cloud in the end of the movie, he says he was finally able to meet Mother. Cloud wonder's what's gonna happen next, but Kadaj doesn't know. "Mother will tell me", he says. Kadaj's journey is over, but with Jenova in the possession of his spirit, Sephiroth's has just begun.
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