"Am I look dashing?" isn't correct. You did not hear it correctly. "Am I looking dashing?" is possible. "Do I look dashing?" is also possible, and more likely. Neither is a common phrase. Both are natural phrases. Someone might say "Am I looking elegant?" "Do I look spiffy?" "Am I looking magnificent?" You can use any word that means "well dressed." In the phrase "Am I looking __________ " you can''t guess what word fills the blank. "Dashing" is an interesting word choice. It means looking very fashionable and elegant. It is a light-hearted, joking thing to say. It suggests that it is surprising or unusual for the person to look that way. Imagine a high school student going to the "prom" (a fancy dance at the end of senior year). He does not own a tuxedo (fancy suit). He rents a tuxedo. He puts it on. His mother might say, "My, don't you look dashing!" It both a compliment and a joke. It means "you don't usually wear that kind of suit" and "you look very well-dressed in that suit." Similarly, a person who had just dressed up in something new and unusually fashionable might say to someone "Do I look dashing?" or "Am I looking dashing?" It is a "rhetorical question." He is expressing false doubt. He knows he looks good.

Again, I am almost certain that you did not hear that correctly. Which drama did you hear this on? And which character said it? Can you give any context?

The phrase is 'Do I look dashing?' It is not an everyday expression. When do you think it is used? What does dashing mean in this case?