Wiktionary
chase after
vb. 1 (&lit chase after English) 2 (context idiomatic English) To pursue someone with romantic intentions; to woo.
WordNet
Usage examples of "chase after".
Little bitty baby Smitty, a voice said way back in his mind, but when he tried to chase after that voice, wanting to ask if it really knew something or was just blowing off its bazoo, it was gone.