That makes sense. So you can walk around and always know your HR, yes?
Alas, the short answer really is: no, there aren't any heart-rate monitor watches that look anything like a "normal" watch. Although some models sort of pass muster. Polar's F4 ($100) has a digital watch face, but it has full watch/calendar functions as well as basic HR and calorie-counting measures. Suunto's t3d ($169) also is sort of tech-y looking but it's no worse than most digital watches out there. And it has day/date/time, plus an alarm.
Perhaps not surprisingly, Timex makes an HR monitor that looks pretty good: the Ironman Race watch ($150). Still digital, but it's a contemporary design that wouldn't be out of place around town.
Myself, I'd probably get something like the t3 for heart-rate use. For a watch to wear, I'd opt for a St. Moritz Pathfinder ($160), a nice-looking, very tough, analog-face watch that has day/date/time plus a rare feature in an analog watch: An alarm. It comes in a variety of face colors and band styles, to suit the look you want.