This is a vibe! The first thing to point out is that "Time 3" and "Time 4" are exact parallel transpositions of "Time 1", which means that yes, basically they "have their own key" as you put it. The other thing (which is what makes this piece such a vibe!) is that each "Time" actually contains two chords, which correspond to when you change the note in the bass. So "Time 1" starts with a A#min, then goes to BMaj when the D# plays in the melody and the B in the bass, then back to A#min when you go back to the C# over the A#. As you say about this not matching standard scales, and as is mentioned in the comments, this generates quite a large "contrast" or what feels like a large "move" in the harmony - the 'standard scales' that work in A#min and the 'standard scales' that work in BMaj don't share many notes - yet it's held together by the bass just rocking between two notes a semitone apart and the melody having strong melodic coherence.
So in terms of wanting to play something over this like an arpeggio etc., if you don't want to add any extra harmonic complexity you'll need to play something that changes to follow this harmonic movement.
As "Time 3" and "Time 4" are parallel to this, we find that "Time 3" moves from C#min to DMaj then back to C#min, and "Time 4" moves from D#min to EMaj then back to D#min.
This is what I consider to be the "fundamental harmony" of what you've played, but as far as the chords you've suggested, absolutely, C#Maj 'works' over "Time 1"/"Time 2" - and would add some extra flavour due to changing the harmonic emphasis - and Cdim 'works' over "Time 3" and Ddim over "Time 4" - but these are chords you could choose to play to add extra sound, and extra harmonic flavour, to what you already have, rather than being "the chord" of the underlying harmony