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 values at points.  Some comments here seem to subscribe to the very common fallacy that the fact that a distribution need not have a value at each point implies that one can never compute its value at any point.  The concept of the value of a distribution at a point was examined in detail by pioneers in the 50's and in fact those distributions which occur in practice tend to have values at most points.  Of course, it is even pre-kindergarten level that the delta distribution (defined anywhere sensible--real line, complex plane, euclidean space, differentiable manifold, fractals: take your pick) vanishes everywhere except at the origin and so your formula holds there (with plus or minus sign as you like).  If you want to consider the value at the origin (which I take to be the main point of your question), then be aware that you are leaving the mainstream.  That isn't necessarily a bad thing but then you have to very precise in specifying  in what sense your concepts and formulae are to be understood, something I consistently miss in your prolific posts.

Finally, the frequent occurrence of the Fourier transform in this thread  seems to me to be the grandaddy of  all red herrings--but that is probably just me.