This talks very much to a single individual's rational, which makes it tricky to answer on WorldBuilding. The rationale for a person is whatever they want. This is doubly tricky because the individual is a god, and they have even more options on the table. However, in this case I think there's some very strong prior art in the Hindu belief system and the god Brahma. The cosmology surrounding Brahma is one of what the philosopher Alan Watts called the three main world views of the world, so it seems general enough to serve a purpose here. (The other two he defines are "the world as a clay pot," shared by the Abrahamic religions, and "the world as nature," shared by the Chinese religions and those affected by them).
This is told per Alan Watt's wording in his lecture series Introduction to Hinduism and some of the related lectures he spoke. I do often wonder how much he adds to the story, but I find the additions good for inspiration. I defer to those who practice the religion if I got any details incorrect (or fundamental errors).
In the Hindu belief, we see Brahma as a non-dual entity with two sides. On one side we have the dreamer, and on the other side we have the dreamed. Thus one can think of Brahama as having one face on one side, and many faces on the other side, one for every character being dreamed -- one for every living entity in existence! They are one in the same.
As Watts describes, the Hindu cosmology one of "the world as a drama." It can be thought of as a world where the writers and the actors and the production team put on a play and got so completely engrossed in it that, for a moment, they forgot it was a play at all. They forgot about the great hall and the auditorium before them. They simply lived an existence, gleefully searching out all the paths for those in the play. Indeed, this is fundamental that the dreamer can "get lost" in his own creation. It is a reason for a god to break into multiple pieces.
According to the end-times story related to this, there comes a great time of strife, where the great demon Kali rises up and takes control of the storyline and all of existence. And in the end, Shiva, in ultimate agony, starts to dance a dance. This is a particular dance, the Rudra Tandava, seen only twice in all of creation. And the stomping of his feet cracks the ground asunder and great flames consume all of creation.
And when the story is over, and all others lie dead, even Kali himself, Shiva looks out at the audience in the great hall, and sees them for the first time. He sees the audience watching him. He sees himself in the audience, for he is an aspect of Brahma, thus he is an aspect of all.
And so, he takes a great bow, and turns to exit at the rear of the stage. And as he turns, we see another face on the back of his head: the face of Brahma, the dreamer. It has been on the back of his head the entire time, dreaming, as it has been on the back of all of our heads. And so the cycle begins again, with Act I, Scene 1, Brahma at the center of the stage. And he begins dreaming again.
Why would a god split into two halves? Any number of reasons. Your imagination is the limit... or not. The answer might not even be limited by your imagination. But if your reason for splitting is to become non-dual: the dreamer and the dream, then drawing upon one of the major classes of cosmology from all of human religion seems to be a decent starting point.
The Archbishop is lying in the grass, asleep. He is dreaming of a grasshopper lying on his chest. Or is he? Perhaps it is the grasshopper that is lying asleep, dreaming of resting on an Archbishop's chest.
Which is real? The Archbishop or the grasshopper?