Pitch shifters do the magic by dividing (shifting left) the frequency spectrum of the original track, so if you want it to sound realistic, you should recreate what is lost in the high frequency part (mostly pick noise and very early part of the note) - you can do it with any "exciter" plugin that creates high frequencies based on what is in the mid and low frequencies (an ordinary eq will not work here).
Then apply your normal bass guitar effects.