Did we get closer to another plane than we were supposed to, or was the pilot just protecting our delicate sensibilities?
From the flightradar24 playback, it appears to have been an encounter with Etihad Airways 171. 

Your flight (Alaska) was at FL360, while the opposing was at FL370. Therefore, there was 1000 feet of vertical separation between them, which is typically considered safe when operating with RVSM, Reduced Vertical Separation Minima rules (see other answers).



I also looked up the LiveATC recordings archive (Seattle Centre Sector 14 high, 28/04 2000Z), and found a clip that I think corresponds to the event:


  
  Alaska 557, can we take a heading right of course to avoid some wind?
  Affirmative
  Ok, deviation approved [...]