  We study the quasinormal mode eigenvalues and eigenfunctions for the
Teukolsky equation in a horizon penetrating, hyperboloidally compactified
(HPHC) coordinate system. Following earlier work by Zengino\u{g}lu
(arXiv:1102.2451), we show that the quasinormal eigenfunctions for the
Teukolsky equation are regular from the black hole horizon to future null
infinity in these coordinates. We then present several example quasinormal
eigenfunction solutions, and study some of their properties in the
near-extremal Kerr limit.
