Of phases and magnitudes

I attended a really interesting talk today by Emmanuel Candes about phase recovery for X-Ray crystallography. While I will not discuss the problem, some thing he mentioned did stick to me. His (fairly believable) claim was that for any signal, the phase is way more important than the magnitude of the signal. 

If you consider a signal as a complex number, it can be represented as: $r \cdot e^{i\varphi}$ where $r$ is the magnitude and $\varphi$ is the phase. For a complex number, intuitively, the phase determines the direction of the signal in the complex plane. If you think of an analogy for high dimensional (say $d$) data, the norm of any point $p \in \mathbb{R}^d$ can be represented by a single value $\left\| p \right\|$  (one dimension) while the direction requires a $d$-dimensional unit vector $\frac{p}{\| p \|}$.

So the norm requires a single dimension while the direction requires all $d$ dimensions (I think it might be possible to represent the direction with one less dimension). So here I can think of two questions:

1. Is there any formal analogy set up between magnitude and phase of signals to the norm and direction of $d$-dimensional points? This question might be stupid in itself
2. Are there any useful applications (real problems) where the magnitude (or equivalently the norm) is more if not as much important as the phase (or direction)?