There are a few ways to go if you're looking for a new fly reel to go along with a new rod: buy a new reel, get a vintage reel, or have one made for you. You'll have to do some thinking & a little research before you decide on the reel you want, but it will be effort well spent in the end. I'll just touch on some of your options here.
Vintage: This is fine on a new custom rod. The proper style & make of reel will go well with a bamboo rod, but don't deny your graphite or fiberglass fly rods either. You'll have to do some research to find the make & model of reel you want. Maybe there's a company who's story & history you like (Grandpa always fished with this make of reel, etc); or maybe there's an older model of reel that just looks like what you've always wanted. You get the idea. Your decision should also be based on both looks & function. When buying a vintage fly reel, it's very important to find a reputable dealer & to ask them any questions you have before you buy.
New: Nothing like starting fresh for the entire outfit of rod & reel. Here, you should be careful of cheap imitators who make their reels look like vintage, collectible reels. "Replicas" are not the same as the real vintage items & you should be careful of reels labeled as such. Some of them are made with very cheap parts with no company to stand behind them.
Instead, look for legitimate reel makers working in the style you like. This could be a single craftsman reel maker or a larger manufacturer, but you need to make sure that they warranty their reels & will stand behind them. I'm not affiliated with any reel makers, but names like Hardy Brothers & Peerless come to mind.
Custom: Just like rod makers, there are reel makers out there who can & will make a reel to your specifications. It usually isn't cheap, but if you want something truly custom this is the way to go. Ask around & search out a competent reel maker to work with, just as you did with rod makers.
When purchasing a new reel, you need to consider style. A large arbor aluminum reel does not traditionally go with a bamboo fly rod. If that's what you really want, fine. Just know & understand the styles of fly tackle. Also a classic style reel will go nicely with any fly rod - bamboo, graphite, or fiberglass. Just make sure after you've found the "look" you like, that you get a reel that will function well & hold up & that fits within your budget.
Finally, make sure the new reel fits your new rod. This is usually nothing to worry about, except when buying some rare vintage reels especially. There are industry standards to the angle of reel feet, but these are rarely followed. Also, you'll want to know the weight of the reel when you're looking around & have an idea of the weight you need to balance out your rod - which, of course, depends on the type & size of fly rod you have.
Common sense & a little research will get you a reel that will round out your new outfit & a reel that you'll be happy to fish with.