If you're planning on holding the soup for any length of time, arrowroot won't hold up. It lasts a few hours if you're lucky. By and large arrowroot is used as a thickener when there's a lot of alcohol or acid -- i.e., when corn starch won't work well; and or when gloss and clarity are very important. But arrowroot is very delicate. Not only will it fail over time, but it can't take boiling.
Corn starch is used as a thickener when clarity and gloss are important and something sturdier than arrowroot is important. Unfortunately, while corn starch will hold up better than arrowroot, it's also fairly fragile over time or high heat.
To my mind, neither is appropriate for the Boston clam chowder we're talking about.
Either use a blond roux, the potato trick, if your chowder is mostly cream you can do a straight reduction. When I make a Boston style clam chowder I start by cooking bacon lardons, setting them aside and leaving the fat in the pan. I sweat onions and leek in the fat, then add a little flour and cook until the raw is off the flour and I have a loose blonde roux. I mash or rice some cooked potato and add that to the roux, then add clam juice, milk and cream, season with salt and white pepper, and bring to a boil only as long as takes for the roux to thicken as much as it's going to -- about two or three minutes. I reduce the heat to a bare simmer, add the cooked, diced potatoes, clams, the reserved bacon lardons, (sometimes some finely chopped sorrel or tarragon), and allow the flavors to marry -- about twenty minutes.
Serve with crackers; saltines, hardtack, or oyster crackers are all good. A good hot sauce, though surely not traditional, goes better than you might think.
The potatoes do most of the thickening, the roux holds the potato/milk binding and keeps things smooth; so portion and proportion the thickeners appropriately. You don't want your chowder too thick, a light nappe is right. If you like a very thick and sturdy soup, crush some crackers and stir them in.
BDL