C

Step 1: Understand the configuration of the electric dipole and the position of the point P.
Step 2: The angle $\theta$ is related to the location of point P with respect to the dipole moment vector.
Step 3: The direction of the electric field $\vec{E}$ created by the dipole can be derived from its potential. For a dipole at the origin, the electric field at a point making an angle $\theta$ with the x-axis involves the dipole angle $\alpha$ as follows.
Step 4: The resulting angle of the electric field with the x-axis can be represented as $\theta + \alpha$.
Therefore, the correct answer is C: $\theta + \alpha$.