Resultant easy axis is perpendiular to film plane for bubbles (or maze structure) stability. Magnetization is along easy axis (normal to the film plane). 

Lok for book O'Dell "Magnetic bubbles"

The maze like structure is a configuration which minimizes the total energy. Such total energy is the sum of different contributions:

-the magnetocrystalline energy, which favours the alignment of the spins along a particular direction (called easy axis)

-the magnetostatic energy, which decreases when the number of domains increases

-the energy stored in the domain Walls, which increases with the number of domains.

-the Zeeman energy, which tends to align the spins along the external applied field.   

The particular maze-like structure you end up with depends on the magnetic history, i.e. how the field was applied (direction, intensity, etc). 

In case this still of interest, consider Book Magnetic Domains: The Analysis of Magnetic Microstructures Springer

Perpendicular (magnetocrystalline) anisotropy is important, because shape anisotropy (dipole-dipole interactions) would normally favor in plane magnetization (creating the magnetic stray field costs energy [cf. "magnetostatic energy" in previous post]).

Domains will form because reversing the magnetic polarization reduces the magnetostatic energy (field energy of the stray field). Their size and shape and configuration will depend on the specific properties of the material, film thickness and "magnetic history", as Jose-Miguel pointed out.