I can think of a few situations in math where a problem becomes easier or an object becomes simpler when some complexity is added. Examples:

1. $S^n$ is never contractible, but $S^{\infty}$ is.

2. The vanishing viscosity method of PDE's.

3. Higher-dimensional topology as opposed to low-dimensional topology (in some specific cases)

4. Singular homology as opposed to simplicial homology

5. Cube complexes as opposed to 3-manifolds

etc.

What other examples are there where a more complex object is simpler to analyze than a 'simpler' object? I realize that you could say that if it is easier to analyze, then it is less complex, so let me restate it this way:

What examples are there where one object seems much more complicated than another, but in fact has a simpler structure?

I've been thinking about things like the Ising model for magnetic phase changes and also about Navier-Stokes; perhaps the simplifications used to derive them make them harder to analyze in the end.