The musician I'd choose is A. R. Rahman, the composer from Chennai. He's written music for hundreds of films in Tamil and Hindi, and also for a few international ones, so most people in India have heard his songs, even if they don't know his name.
His music is hard to put into one category. He mixes classical Carnatic melodies with electronic beats, Sufi singing, Western orchestras and sometimes even sounds from everyday life, like rain or traffic. One song might start with a single flute and end up with a full choir. Even though it's mainly film music, a lot of his songs work perfectly well on their own.
I first came across his work through my mother, who used to play his early film soundtracks on cassette while she cooked. I didn't pay much attention as a child, but when I was about sixteen I heard one of those songs again, played by a street band during a festival, and I suddenly realised how layered it was. After that I went back and listened to almost everything he'd made.
There are several reasons I like him. Musically, I love that every time I listen to a song, I notice something new, perhaps a quiet instrument underneath the main tune. I also like the fact that he's never stuck to one style; he keeps experimenting even though he's already very successful. And personally, his music is tied to my family. Whenever I'm homesick here in Toronto, where I'm doing my master's, I put on one of those old soundtracks and I can almost smell my mother's kitchen. I don't think many artists can do that for someone.