Of course for the standard configuration(i.e. two opposite charges separated by some distance), there exist higher order multipole moments. However for an ideal dipole(those dipoles associated with point mass, which means the separation is zero), are there any higher order moments?
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1No. The contributings of a pure dipole are purely dipole. – David H Oct 29 '13 at 06:31
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4A pure dipole is defined by the condition that its multipole expansion has only a dipole term, so the the answer is no by definition. – joshphysics Oct 29 '13 at 07:51
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Even a single point-like monopole displaced from the origin will have higher multipoles if we expand it's field with respect to the origin, see this question! The same is valid for the dipole. However, a point-like dipole at the origin has a trivial expansion with respect to the same origin. Therefore, in order to have an expansion that is as simple as possible (namely, where the first terms are possibly zero), you should consider to expand your distribution around its center of charge.
