There are two aspects of this question.

1.  Is the delta distribution (on the plane) symmetric with  respect to complex conjugation? The answer is, of course, a resounding yes.  It is even symmetric with respect to reflection in any line through the origin, or any rotation around the origin, indeed under the action of any diffeomorphism of the plane which leaves the origin and satisfies the obvious scaling condition on its derivative there.  This is kindergarten level in the theory of distributions.

2. The second (implicit) question is whether this symmetry can be expressed pointwise, i.e. in terms of its values at points.