Well there's many ways to analyze contemporary pieces, but you can't look at them like you would look at say, Brahms. i.e. "Oh look that's a V going to I." Generally, composers now a days don't really follow functional harmony. The piece you posted makes use of a lot of chordal extensions, i.e. add 2s, major 7ths, etc. If I were looking at a contemporary piece, I would look to see if there was any predominating intervallic patterns both harmonically and melodically. Does the melody make use of big leaps ( it does in this piece)? Are the chords triadic, clusters, fourths, or is it more contrapuntal? How does the composer transform and vary the initial motifs/themes throughout the piece? What's the overall form? Does it even have a clear cut form? These are just some questions you could ask yourself to get started.