My wife and I have a disagreement on which term is the more correct.
Google has references for both. Wikipedia has a page for only one.
I have always heard it as the witching hour. She the other. I told her that she must be referring to a local after school segment on one of her local television stations that ran back-to-back reruns of Bewitched.
So which is it? 
Witching hour. While one can be bewitched (as well as bothered and bewildered) it’s a different thing. The problem stems from similar sounding words being misused by those who are not detail oriented/pedantic/careful/anal retentive about language. ETA: Which isn’t to say that she isn’t careful about proper usage, but that she grew up listening to people who weren’t.
[QUOTE=Hamlet, Prince of Denmark]
Tis now the very witching time of night,
When churchyards yawn and hell itself breathes out
Contagion to this world: now could I drink hot blood,
And do such bitter business as the day
Would quake to look on.
[/QUOTE]
Definitely “witching hour”.
Your cite actually makes the counter-argument, since it’s a poetic text in pentameter. It is possible that Shakesy elided “bewitching” in order to fit the meter, and that the common phrase “witching hour” in speech derives from the single well-known source beside the fuller form, making both correct. The OED says that all usages of the word to modify time follow Shakespeare.
The only unambiguously pre-Shakespearean source for “witching” in the OED is Reginald Scot’s Discoverie of Witchcraft, from 1584, modifying “pen” and “words.” OED is less generous with its citations of “bewitching,” but its first attestanting modifying time is only 1827 (and the first attestation altogether from 1533).
I’d say that both are valid, but since the phrase is an allusion to Hamlet the poetic form is the more correct.