I have a broken hydraulic caliper and am wondering if it can be fixed? Can I buy the half that the broken mount is on or would it be better just to buy a new one?
I highly doubt you'll be able to find a separate alloy casting to replace the one that broke. Even if you can you'll be faced with assembling the caliper which is not trivial.
Just buy a replacement caliper. Make sure you refill/bleed the hydraulic fluid according to Shimano's documented procedure, of have a competent bike shop do it for you.
It is a brake - a critical safety component. Just replace it.
Any fix or hack or bodge could fail you at the worst moment, endangering you and others around you. This one has already broken, who's to say the rest of the casting is any better?
If it helps the acceptance process, find the cheapest brake caliper that you would accept. This is your Sunk cost, what you HAVE to spend. Then subtract that sunk cost from whatever the ideal brake's price is, and that's your UPGRADE cost, so the amount you choose to spend to get something fancier/better/etc. A $100 brake is really only costing you $40 more than the minimum $60 brake. (self-deluding mind games, also useful on any non-cyclist life-partners)
Just replace it.
If money is an issue, it may be reasonable to buy a used brake from ebay/gumtree/craigslist/trademe or whatever. But additional risks include damage from previous owner - you don't know what's happened to it.
For piece of mind I'd go with a new caliper, doubly so on the front where 90% of your braking happens.