Watch the Fingerpicking Pattern 1 online guitar lesson by Muriel Anderson from 50 Right-Hand Techniques You Must Know
Changing the alternation between bass strings gives you more of a professional sound. For example, on a G chord I play strings 6, 2, 4, 3, 5, 2, 4, 3 and on a C chord I'll play, 5, 2, 4, 3, 6, 2, 4, 3, and move my left hand finger from the 5th to the 6th string (3rd fret) when I play the 6th string.
The little example you recognize is a bit of the melody of "Blowing in the Wind." If you're interested in developing the ability to keep the alternating bass going and play melody notes at the same time, you might check out my DVD/book "Building Guitar Arrangements From the Ground Up" released some years ago.