Determine the locus of points $M$ in the plane of a given rhombus $ABCD$ such that $MA\cdot MC+MB\cdot MD=AB^2$.
1. **Define the vertices of the rhombus in the Cartesian coordinate system:**
   - Let the vertices of the rhombus be \( A(a, 0) \), \( B(0, b) \), \( C(-a, 0) \), and \( D(0, -b) \).
   - The center of the rhombus is the origin \( O(0, 0) \).

2. **Express the points as complex numbers:**
   - \( z_A = a \)
   - \( z_B = bi \)
   - \( z_C = -a \)
   - \( z_D = -bi \)
   - \( z_M = x + yi \) for the variable point \( M \).

3. **Calculate the distances \( MA \), \( MB \), \( MC \), and \( MD \) using the modulus of complex numbers:**
   \[
   MA = |z_M - z_A| = |x + yi - a| = \sqrt{(x - a)^2 + y^2}
   \]
   \[
   MB = |z_M - z_B| = |x + yi - bi| = \sqrt{x^2 + (y - b)^2}
   \]
   \[
   MC = |z_M - z_C| = |x + yi + a| = \sqrt{(x + a)^2 + y^2}
   \]
   \[
   MD = |z_M - z_D| = |x + yi + bi| = \sqrt{x^2 + (y + b)^2}
   \]

4. **Express the given condition \( MA \cdot MC + MB \cdot MD = AB^2 \):**
   - Since \( AB = \sqrt{a^2 + b^2} \), we have \( AB^2 = a^2 + b^2 \).

5. **Rewrite the condition using complex numbers:**
   \[
   MA \cdot MC + MB \cdot MD = |z_M - z_A| \cdot |z_M - z_C| + |z_M - z_B| \cdot |z_M - z_D|
   \]
   \[
   = |(x + yi - a)(x + yi + a)| + |(x + yi - bi)(x + yi + bi)|
   \]
   \[
   = |(x^2 + y^2 - a^2)| + |(x^2 + y^2 + b^2)|
   \]

6. **Simplify the expression:**
   \[
   = |z_M^2 - a^2| + |z_M^2 + b^2|
   \]
   \[
   = |a^2 - z_M^2| + |z_M^2 + b^2|
   \]

7. **Analyze the equality condition:**
   - The equality \( |a^2 - z_M^2| + |z_M^2 + b^2| = a^2 + b^2 \) holds if and only if:
     \[
     a^2 - z_M^2 = k(z_M^2 + b^2) \quad \text{or} \quad z_M^2 + b^2 = k(a^2 - z_M^2)
     \]
     where \( k \ge 0 \).

8. **Case \( k = 0 \):**
   - If \( k = 0 \), then \( a^2 - z_M^2 = 0 \) or \( z_M^2 + b^2 = 0 \).
   - This implies \( z_M^2 = a^2 \) or \( z_M^2 = -b^2 \).
   - Therefore, \( M \) is one of the vertices \( A, B, C, D \).

9. **Case \( k > 0 \):**
   - For \( a^2 - z_M^2 = k(z_M^2 + b^2) \):
     \[
     z_M^2 = \frac{a^2 - kb^2}{k + 1}
     \]
     - If \( a^2 - kb^2 \ge 0 \):
       \[
       z_M = \pm \sqrt{\frac{a^2 - kb^2}{k + 1}} \in \mathbb{R}, \quad |z_M| < a
       \]
       - Let \( u \in [0, a) \):
         \[
         \sqrt{\frac{a^2 - kb^2}{k + 1}} = u \implies k = \frac{a^2 - u^2}{b^2 + u^2} > 0
         \]
         - Thus, \( M \in AC \).

     - If \( a^2 - kb^2 \le 0 \):
       \[
       z_M = \pm i \sqrt{\frac{kb^2 - a^2}{k + 1}} \in \mathbb{C} \setminus \mathbb{R}, \quad |z_M| < b
       \]
       - Let \( v \in [0, b) \):
         \[
         \sqrt{\frac{kb^2 - a^2}{k + 1}} = v \implies k = \frac{a^2 + v^2}{b^2 - v^2} > 0
         \]
         - Thus, \( M \in BD \).

10. **Combine the results:**
    - The locus of the point \( M \) is the union of the diagonals of the rhombus, including the vertices.

The final answer is \( \boxed{ \mathcal{L}_M = [AC] \cup [BD] } \).