Let $A, B, C$ and $D$ be four points in general position, and $\omega$ be a circle passing through $B$ and $C$. A point $P$ moves along $\omega$. Let $Q$ be the common point of circles $\odot (ABP)$ and $\odot (PCD)$ distinct from $P$. Find the locus of points $Q$.
1. **Identify the given elements and their properties:**
   - Points \( A, B, C, D \) are in general position.
   - Circle \(\omega\) passes through points \( B \) and \( C \).
   - Point \( P \) moves along \(\omega\).
   - Circle \(\odot(ABP)\) passes through points \( A, B, P \).
   - Circle \(\odot(PCD)\) passes through points \( P, C, D \).
   - Point \( Q \) is the common point of circles \(\odot(ABP)\) and \(\odot(PCD)\) distinct from \( P \).

2. **Analyze the properties of point \( Q \):**
   - Since \( Q \) lies on \(\odot(ABP)\), the power of point \( Q \) with respect to this circle is:
     \[
     QA \cdot QB = QP \cdot QP
     \]
   - Similarly, since \( Q \) lies on \(\odot(PCD)\), the power of point \( Q \) with respect to this circle is:
     \[
     QC \cdot QD = QP \cdot QP
     \]

3. **Use the Radical Axis Theorem:**
   - The radical axis of two circles is the locus of points that have equal power with respect to both circles.
   - The radical axis of \(\odot(ABP)\) and \(\odot(PCD)\) is the line on which \( Q \) lies.

4. **Determine the locus of \( Q \):**
   - Since \( P \) moves along \(\omega\), the circles \(\odot(ABP)\) and \(\odot(PCD)\) change dynamically.
   - However, the radical axis of these circles remains fixed as \( P \) moves.
   - The radical axis of \(\odot(ABP)\) and \(\odot(PCD)\) is a line that passes through the fixed points \( A \) and \( D \).

5. **Conclusion:**
   - The locus of \( Q \) is a circle passing through the fixed points \( A \) and \( D \).

The final answer is \(\boxed{\text{The locus of points } Q \text{ is a circle passing through } A \text{ and } D.}\)