Is an expression of a quadrupole as an expansion of dipoles possible? Would it be possible to express a quadrupole as an expansion of dipoles? Because a possible definition of a quadrupole seems to be: an electric field equivalent to that produced by two electric dipoles.
 A: You seem to mix up two things here: Which charge distributions generate what multipole fields and how multipole-expansion works.
Yes, you can reach a pure quadrupole field by a limiting procedure involving two dipoles (just take two anti-parallel dipole sources and let their distance go to zero from above).
But that doesn't mean you can expand the quadrupole field into dipoles in the sense of a multipole expansion. One natural way to think about multipole expansion is in the sense of a far-field approximation: If you have an arbitrary configuration of charges restricted in a radius $R$ around the origin, then the field for $r \gg R$ will be given by a multipole expansion, the leading term will be the lowest non-vanishing multipole contribution.
If you have a quadrupole (constructed as above), you will see that the dipole moment is zero (because the dipole moments of the two dipoles cancel out). In a multipole expansion there is no way to put two dipoles "next to each other", all the multipole sources in the expansion are exactly located at the origin.
A: Actually, for any kind of magnetic field or electric field when a multipole expansion is done in different poles, as dipole, quadrupole, sextupole, octupole and so on, it is like a Taylor expansion:
$$ B_z(x) = \overbrace{B_{z0}}^{dipole} +\overbrace{\frac{dB_z}{dx} x}^{quadrupole} + \frac{1}{2!} \overbrace{\frac{d^2 B_z}{dx^2} x^2}^{sextupole} + \ldots$$
one observes that a dipole field is constant whereas a quadrupolar field depends linearly on the offset which the particle has with respect to the origin. From this point of view a quadrupole cannot be made of 2 dipoles.
Actually, you could put 2 dipoles one having an offset $R$ the other one having an offset $-R$ laterally to the origin. Assume that the distance $2R$ is just large enough to have some lateral space inbetween. Then you would rely on that the stray field between the 2 dipoles makes up a quadrupole.
The following comment has to be made in order to avoid confusion.
Actually a dipole with a stray field would not be a dipole in the strict sense, because a dipole in the strict sense has just constant field and would not allow for a stray field.
Of course a realistic dipole would never be a dipole in the strict sense and can (and will) have some stray field.
Probably you would get some quadrupolar component of these combined stray fields, but this combination field would actually be mixture of many different terms in the multipole development that one would never call that a quadrupole.
