What is an effective metric of complexity for an Artificial Neural Network?
The standard complexity metric in theoretical computer science and machine learning, in particular in statistical learning theory, is the Vapnik–Chervonenkis (VC) dimension. It is of interest because it gives us a very good tool to measure the learning ability of a neural network (or any other statistical learner, in general).

A good introduction to the use of VC dimension for studying neural nets is:

Eduardo D. Sontag [1998] "VC dimension of neural networks" [pdf].

There, the author shows (for instance) that a network with one hidden layer, $n$ inputs, and $\tanh$ neurons has VC dimension of $n + 1$. He also explain some basic technique for how to upper-bound the VC dimension, and for how to use it for dynamic neural nets.