AA, is it my imagination, or is Koine
 

a dumbed down version of Greek?

Solon, chime in. It seems the tenses and declinations are not as complex. But I'm just fumbling around self-taught, so I could be off-base.

I note Homer wanders around a lot and it requires reading it with grammar, whereas the NT doesn't seem quite as complex.

You've hit the nail on the head. Koine is simplified Greek, since it was pretty much everyone's second language. It's mainly aorists, with relatively short sentences and straightforward sentence structure, which makes it nice and easy to read.

I concur. It is actually possible to work your way through the gospels as a beginner. New Testament Greek is significantly simpler than Homeric or even Attic Greek.