Researching an article that pours complimentary drinks as a matter of policy. For instance, here in Colorado, Boulder's Frasca follows the Friulian tradition of the tajut. A fellow 'hound also told me about a place in Phoenix that serves gratis glasses of Sambuca after the meal.
I assume most places that do this will be places that emphasize prix-fixe tasting menus and such. In short, the price of the drink, be it an aperitif or a sparkler or a digestif, is technically probably covered by the price of the meal. Even so, if you know of places that do this anywhere in the country, I'd be much obliged to hear about it.
A few of us chowhounders dined at a new French Restaurant - 'Quatrefoil' outside of Toronto a few days ago. ( chefs/owners honed their skills under two of the most demanding Michelin3* chefs, Gordon Ramsay and Marco Pierre White ).
We had the chef's tasting menu. At the end of the meal we were offered an additional complimentary chocolate dessert that was paired with a glass of complimentary ' 10 yrs old Tawny Port '.
My (essentially bottomless) glass of Nuits-St-Georges at Guy Savoy (Paris) was complimentary. Then again I was having the Couleurs, Textures et Saveurs menu that was going along with the Leflaive Les Pucelles Puligny-Montrachet.
I've had the odd complimentary pours at restaurants here and there, but there's really been no consistency.
The Crystal Lake Hotel in the Poconos, which serves high-end italian, routinely brings snifters of Sambuca romana at the end of a meal, always with 3 coffee beans for good luck.
Recently at La Calle Doce, a high end Mexican resto in Dallas, our group was surprised by a complimentary round of peach liqueur cocktails with our flan, a nice touch.
Neither is a prix-fixe format, but I think the size of the bill is a factor.
Chez Panisse in Berkely offers an aparatif as part of their Saturday prix fixe
Marica in Oakland, Ca offers a glass of sparkling wine if they know you are a first time diner. Actually they give out sparkling wine on the house for all sorts of reasons. It is a regular menu and not prix-fix or tasting.