What is the best continuation in this problem-h4 or Qd8? In the given diagram, I have two best ways to continue for white, which is the better option out of these two and why?

[FEN "2Q5/p1P2pkp/1p2p3/5p2/3P4/5P1P/P2q2PK/8 w - - 0 1 "]

1. Qd8 Qf4+ (1. h4 Qf4+ 2. Kh3 Qc1 3. Qd8 Qh1+ 4. Kg3 Qe1+ 5. Kh2! f4 6. Qg5+ Kf8 7. c8=Q#)2. Kg1 Qe3+ 3. Kf1 Qc1+ 4. Kf2 Qd2+ 5. Kg3 f4+ 6. Kh4 Qe1+ ( 6. Kg4 Qxg2 7. Kh4 Qg3+ 8. Kh5 Qg6+ ½-½) 7. Kh5 Qe5+ 8. Qg5+ Qxg5 9. Kxg5 1-0


Also would be interesting to see what line Houdini selects and how much will be the points for each of these two moves...

problem taken from here
4...Qd2+ in your main line is an error; instead Black should play 4...f4, after which he delivers perpetual check. White can try to play on with 5.Qg5+ Kf8 6.Qxf4 Qxf4 7.c8Q+ but that's still a draw. So 1.Qd8 does not work.

1.h4 enables White to escape from perpetual check, and wins.