Let $n$ be a positive integer and let $P$ be the set of monic polynomials of degree $n$ with complex coefficients. Find the value of 
\[ \min_{p \in P} \left \{ \max_{|z| = 1} |p(z)| \right \} \]
1. **Define the problem and the set \( P \):**
   Let \( n \) be a positive integer and let \( P \) be the set of monic polynomials of degree \( n \) with complex coefficients. We need to find the value of 
   \[
   \min_{p \in P} \left \{ \max_{|z| = 1} |p(z)| \right \}.
   \]

2. **Consider a specific polynomial \( p(z) = z^n \):**
   For the polynomial \( p(z) = z^n \), we have:
   \[
   \max_{|z| = 1} |p(z)| = \max_{|z| = 1} |z^n| = 1.
   \]
   This shows that the maximum modulus of \( p(z) \) on the unit circle is 1.

3. **Show that \( \max_{|z| = 1} |p(z)| \ge 1 \) for any \( p \in P \):**
   Suppose, for the sake of contradiction, that there exists a polynomial \( p \in P \) such that:
   \[
   \max_{|z| = 1} |p(z)| < 1.
   \]
   This implies that for all \( z \) with \( |z| = 1 \), we have:
   \[
   |p(z)| < 1.
   \]

4. **Apply Rouché's theorem:**
   Consider the polynomial \( z^n \) and \( p(z) \). By Rouché's theorem, if \( |z^n - p(z)| < |z^n| \) on the unit circle \( |z| = 1 \), then \( z^n \) and \( z^n - p(z) \) have the same number of zeros inside the unit disk.

   Since \( |p(z)| < 1 \) for \( |z| = 1 \), we have:
   \[
   |z^n - p(z)| \ge |z^n| - |p(z)| = 1 - |p(z)| > 0.
   \]
   This implies that \( z^n - p(z) \) has \( n \) roots inside the unit disk.

5. **Contradiction:**
   However, \( z^n - p(z) \) is a polynomial of degree \( n \), and if \( |p(z)| < 1 \) for all \( |z| = 1 \), then \( z^n - p(z) \) would have \( n \) roots inside the unit disk, which contradicts the fact that \( z^n - p(z) \) is a polynomial of degree \( n \) and cannot have more than \( n-1 \) roots inside the unit disk.

6. **Conclusion:**
   Therefore, our assumption that there exists a polynomial \( p \in P \) such that \( \max_{|z| = 1} |p(z)| < 1 \) is false. Hence, for every \( p \in P \), we have:
   \[
   \max_{|z| = 1} |p(z)| \ge 1.
   \]

Thus, the minimum value of \( \max_{|z| = 1} |p(z)| \) for \( p \in P \) is indeed \( 1 \).

The final answer is \( \boxed{ 1 } \).