This is a very broad question, the answer to which largely depends on the composer and the end result they are aiming for. Given your clarification, I can provide some directions that I would personally go for.
First of all, you should determine the key the piece is in. This particular one sounds like E minor to me (I don't have perfect pitch, so correct me if I'm wrong).
There are many extending techniques when it comes to making covers and composing your own tracks.
For "traditional" piano (left hand plays chords, right hand plays melody) I would follow this algorithm:
- Write down the chord progression for the track
- See if you can invert some chords or dilute the progressions to make it more "interesting"
I am assuming you are familiar with chord inversions, so I will not explain the concept here. I am going to give you a list of how chords can be substituted ("diluted") instead:
- Go into a relative key (E minor → G major) and borrow chords therefrom
- Go into a parallel key (E minor → E major here)
- Play around with chords that have the same root (like Esus here instead of Em)
- Play around with chords that have similar notes (similar to using chord inversions) and omit notes therefrom if they don't fit the key
- Use (common) chord progressions instead of playing the same chord bar after bar (an example would be playing, say, Em-F#dim-Bsus-Em instead of Em-Em-Em-Em)
A more "complex" way to approach the problem would be to compose a counterpoint instead of playing chords. You could construct one by using "chord theory" or by simply developing a countermelody for the left hand. This is a more "destructive" approach if you are trying to make a cover, even more destructive than using "chord dilution". It might make the initial piece unrecognisable, but it all depends on what you're aiming for.
If you are familiar with blues progressions (and lazy like me), you can use blues chord substitutions for most "rock" compositions. Just run a search for blues chord matrices. I usually use 16-bar ones. The regular matrix for 16-bar blues is I-I-I-I | IV-IV-I-I | V-IV-V-IV | V-IV-I-I. There are plenty of options out there to create transitions and substitute chords within this matrix. Just compare the different ones and see which chords and progressions can be exchanged with other ones and apply them to your composition.