Compartmentalization of the Eye, then Extreme Specialization within each Compartment.
Let's talk about human eyes for a moment here. Humans use one type of eye to do everything. Technically, yes, we have two eyes, but from a biological perspective, they're more or less identical in function, and they only operate in a pair to give humans depth perceptions. Because of this, the human eye is designed to be super versatile. It has a lenses which can adjust itself to focus between objects which are closer or farther away. It also lies within an occipital socket that allows the eye to move and focus on various things within it's field of vision. Note that the occipital socket doesn't actually give you a larger field of vision (a marginal improvement at best), just the ability to focus on details at any range. In other words, since humans have only had one type of eye to use, the eye can do everything. This however, comes with a drawback - jack of all trades, master of none. Trade-offs exist, and since the human eye is designed to be able to do everything, it can't do everything as well as an eye specialized for the task can. It also usually comes with a defect of some kind in certain areas, which is why most people wear some type of correctional lenses or another, i.e. glasses or contact lenses.
Various other types of animals have eyes which are specialized for certain tasks, but are absolutely useless when it comes to others. 'Eagle eyes' are a good example here - they cannot rotate their eyes, they're not so good at seeing color (even though they can see more colors than humans can), and they have a blind spot directly in front of them. Not something that I'd ever actually want.
However, the insect's unique compound eyes present a way to get everything, and that's by having patches of the eye each be uniquely adapted to whatever the insects needs. In fact, this already occurs among some insects, like superior flier such as dragonflies, where they have specialized section for acute vision. Essentially, each part of the compound eye would be claimed by whatever part needed it the most. It would have a section devoted to long distance acute vision to spot things far away, and a section devoted to seeing things up close to avoid the blindspot. (Kind of like bifocal glasses, funnily enough.) Most of what it would see would actually be in grayscale, given that seeing in color would take up precious space in the eye, but sections of it would evolve to pick out and see colors with precision, possibly even a greater range than we humans have. The eyes on the extreme edge would be made not for accuracy, but rather for extreme angles so they could see as far around themselves as possible.
The one thing I'm not sure about is whether or their eyes could move. It's possible that they would eventually develop an occipital socket to move their eye around, but it's more likely that with how complex their eyes become, they wouldn't be able to move them and would need to turn their whole head to see things with the relevant part of their eyes.