Signal processing - why compute imaginary part?
One use of Hilbert Transform is to recover the amplitude envelope of a signal.

Here is a practical example: Extracting Binary Magnetic-Strip Card Data from raw WAV

Often the shortest distance between two points is through complex analysis.  Even if the start and destination don't appear to involve complex numbers.

Regarding "What is the phase shift of 90°?": If it is single cisoid, i.e. a single point making a circular orbit around the origin on the complex plane, then applying a phase shift of 90° obviously sets it backwards or forwards by a quarter-cycle.  Now if you consider an arbitrary waveform to be a sum of cisoids with different amplitude, phase and frequency, (Fourier Theory shows that you can generally achieve this), you can just set each one back by a quarter cycle, and recombine. That's one way to accomplish the Hilbert Transform.

http://bl.ocks.org/jinroh/7524988 &lt;-- this gives a visual example of combining cisoids into the target waveform.