Can this recipe be hot water processed to safely can/preserve? If so, how long should it be processed?

I don't think this compote is a good candidate for canning, if you are hoping to keep it as a compote. You could preserve the flavors in the form of a jam or jelly, though. The pomegranate seeds and raspberries will break up in the cooking process, and leave you with bits that could be rather unpleasant chewing and bitter tasting. I'd suggest freezing as an alternative method of preservation, which won't require any more heating be done. There will be minor changes in texture, and it may thaw a bit thinner as a result- but you could thicken it as desired at that point. They make jam jars for freezing too - so you can store it in smaller quantity containers. I imagine it would keep frozen for several months just fine.