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Just got the Alesis Monitor One Mk2s [Passive not Active] and I'm wondering does the amp have any effect on the quality of the sound?

Yes

Shall I sell these and get the M1 Active Mk2s or just buy a good amp, recommend me an amp if the latter please.

Depends on your budget and space.
Better amps will produce better signal than built in amps in less expensive active monitors.
However you have to run cabling which is prone to interference.
And you need more space now for an amp to drive your monitors.
Further, well designed (expensive) active monitors will have design advantages due to proximity of amplifying circuitry and speakers so will generally produce more accurate results than a more expensive speaker/amp combination.

Go look at studio amps. Much like monitors, price is generally proportional to quality. sound on sound has great gear reviews.
The nice part about going passive is that you can upgrade your monitors later and keep your amplifying circuitry.

Most people these days rock active gear.

Thanks for the tips I'll look into it.