Yes, Hadith are also inspiration from Allah :swt:. Allah :swt: Himself says in the Quran about the Prophet :saws:
Nor does he speak from [his own] inclination.
It is not but a revelation revealed, [Qur'an 53:3-4]
Yes, the Qur'an is complete, and to understand the Qur'an, you need the Hadith and Sunnah. Allah :swt: says about the Prophet :saws:
Just as We have sent among you a messenger from yourselves reciting to you Our verses and purifying you and teaching you the Book and wisdom and teaching you that which you did not know. [Quran 2:151]
Allah :swt: also says:
It is He who has sent among the unlettered a Messenger from themselves reciting to them His verses and purifying them and teaching them the Book and wisdom - although they were before in clear error - [Quran 62:2]
Qur'an has the verses, but what about explanation about purification and wisdom and knowledge of the things that we did not know? All these were also taught by Prophet :saws: as instructed by Allah :swt: and without Hadith, they will be lost.
Mufti Taqi Uthmani explains this question by saying:
The answer to this question is found in the Holy Qur'ân itself. A combined study of the relevant verses reveals that the Holy Qur'ân deals with two different types of subjects. One is concerned with the general statements about the simple realities, and it includes the historic events relating to the former prophets and their nations, the statement of Allâh's bounties on mankind, the creation of the heavens and the earth, the cosmological signs of the divine power and wisdom, the pleasures of the Paradise and the torture of the Hell, and subjects of similar nature.
The other type of subjects consists of the imperatives of
Sharî'ah, the provisions of Islâmic law, the details of doctrinal issues, the wisdom of certain injunctions and other academic subjects.
The first type of subject, which is termed in the Holy Qur'ân as
Zikr (the lesson, the sermon, the advice) is, no doubt, so easy to understand that even an illiterate rustic can benefit from it without having recourse to anyone else. It is in this type of subjects that the Holy Qur'ân says:
And surely We have made the Qur'ân easy for Zikr (getting a lesson) so is there anyone to get a lesson? (54:22)
The words "for
Zikr" (getting a lesson) signify that the easiness of the Holy Qur'ân relates to the subjects of the first nature. The basic thrust of the verse is on getting lesson from the Qur'ân and its being easy for this purpose only. But by no means the proposition can be extended to the inference of legal rules and the interpretation of the legal and doctrinal provisions contained in the Book. Had the interpretation of even this type of subjects been open to everybody irrespective of the volume of his learning, the Holy Qur'ân would have not entrusted the Holy Prophet with the functions of "teaching" and "explaining" the Book. The verses quoted earlier, which introduce the Holy Prophet as the one who "teaches" and "explains" the Holy Qur'ân, are explicit on the point that the Book needs some messenger to teach and interpret it. Regarding the type of verses which require explanation, the Holy Qur'ân itself says,
And these similitudes We mention before the people. And nobody understands them except the learned. (29:43)
Thus, the "easiness" of the subjects of the first type does not exclude the necessity of a prophet who can explain all the legal and practical implications of the imperatives contained in the Holy Qur'ân. (Taqi Usmani,
The Authority of Sunnah, Chapter 2: The Scope of the Prophetic Authority,
Source)
Read more on the refutations to Hadith-rejectors claims by following the links posted here:
http://www.islamicboard.com/hadeeth/45363-hadeeth-rejectors-refutations.html