Fractal dimension of a fractal determines its complexity. (What does this mean?)

This was a tough one. I feel like everything that has to do with fractals is very ambiguously defined. Even the definition of a fractal seems to be up in the air, defined with more, or less parameters depending on who or what your source is.

Upon first look at the statement,"the fractal dimension of an object determines its complexity" I would have to think that the fractal dimension has something to do with how intricate the rendering of the fractal is from a visual standpoint. When I think of the word complex it leads me to thinking that something looks confusing, or is dense with information. Thus, I believe the actual confusion lies on the word, complex. What does it mean?

The Oxford Dictionary defines something as "complex" if it "is made of closely connected parts". This could mean that something would be more complex if it had more closely connected parts. I believe this gives us a good realization of what the "complexity of a fractal" could mean.