Why did Prophet Muhammed need to appoint a successor or Imam in the first place?

The underlying assumption of this question is that the Qur'an as it exists today, in the form of a physical text or book, is sufficient for the guidance of mankind in all matters. However, Muhammad did not present, produce, or leave a book or a text. The Qur’an was purely oral and it was not a volume text  – rather, every Qur’anic recitation or set of recitations was revealed through Muhammad to address very specific situations and events, and when events and new situations arose, the Prophet recited new recitations to respond and guide the community for that new situation and it continued this way until he died.

If this point is grasped and when it is realized that the Qur’an in its original context and form was nothing like a book or a scripture intended to provide guidance indefinitely into the future (if it were intended as such, the Prophet would have compiled it into a book or God would have sent it as a complete book), then the question of a Successor to Muhammad becomes an imperative.

Muhammad needs a successor for the following reasons:

1. Qur'anic Norms and Precedent: Prophetic family succession is an established principle in the Quran and according to S. M. Jafri, the number of verses emphasizing the special status of a Prophet’s family numbers to over 100. Based on the Quranic model, it is not only reasonable but obvious that Prophet Muhammad’s family, his Ahl al-Bayt, holds a special and sanctified status among all the believers – and this is also consistent with the Arabian concept of Ahl al-Bayt. Thus, it would be entirely natural if not axiomatic that Prophet Muhammad appointed his successor by Divine inspiration from among his Ahl al-Bayt just as prior Prophets did like Abraham, Ishmael, Isaac, Jacob, David, and Zakariyyah had done per the Qur’an. Several academic scholars of early Islam have analyzed the Qur’an as a historical and theological document and have concluded that the Qur’anic model of Prophetic succession is one where God always appoints heirs and successors to a Prophet from the family members and descendants of that Prophet and these descendants of Prophets are granted both Prophethood, the Kitab (a symbol for continuous Divine Guidance) and Hukm (theoretical and practical wisdom to judge) - see 4:54, 57:26, 29:27, 6:84–89. When prior Prophets have successors (who are not always new prophets), then Prophet Muhammad should also have a successor. For specific cases mentioned in the Qur’an, see this article.
2. Practical necessity: all power and authority was held by Muhammad during his mission - religious, moral, military, economic, social, etc. Muhammad per the various Medinah verses of the Qur’an was the undisputed deputy of God’s will, the judge of all matters on God’s behalf, the conduit of Divine Guidance, and the intercessory mediator between God and humankind in matters of forgiveness and repentance and purification. It is obvious that Muhammad NOT appointing a successor would lead to chaos and a massive power struggle by various parties. Old tribal rivalries, feuds, politicking, and power grabs that existed prior to Muhammad uniting all these tribes would come back instantly.
3. Divine Justice and Logical Necessity: Prophet Muhammad’s role was NOT to simply deliver a text-book called the Qur’an to his people and then leave them to it. This book-only model of divine guidance is NOT Qur’anic. In fact, in several places the Qur’an explicitly REJECTS the “book delivery” of Divine Guidance as something false – see 6:7, 4:153. Instead, the Qur’anic model of Divine Guidance is one where the Prophet BOTH recites divinely-inspired recitations (qur’ans) on a piecemeal and responsive basis AND ALSO provides continuous guidance to the community on all matters as and when they come up - see the verses 2:151, 62:2, 3:164, 16:44. To pose a question in the form of an analogy, would it be fair or logical that the community of believers during the time of the Prophet had Google Maps for guidance but the communities to come in later times only had access to a paper map that kept getting older with time and becoming less and less reflective of the situations of their own time? This is not to say that there are not timeless truths, morals, and ethics in the Qur'an but not all guidance was general and effective for all time. As stated earlier, every Qur’anic recitation or set of recitations was revealed through Muhammad to address very specific situations and events as they arose in real time.

Explication of the divine intention of the revelation was among the functions that the Qur’an assigned to the Prophet. The Prophet functioned as the projection of the divine message embodied in the Qur’an. He was the living commentary of the Qur’an, inextricably related to the revelatory text. Without the Prophet the Qur’an was incomprehensible, just as without the Qur’an the Prophet was no prophet at all.

– Abdul Aziz Sachedina, (“Scriptural Reasoning in Islam”, Journal of Scriptural Reasoning, 5/1 2005)

Basically, the Prophet held an indisputable and invaluable set of additional religious and spiritual functions  in addition to reciting the Qur’an and the logic of Divine Justice necessitates that someone similar to the Prophet remains in the world after him to continue performing all the spiritual functions which were essential to the early community and must continue to be fulfilled for all times. See this article.