What can move both as a longitudinal wave and a transverse wave at the same time?

Water waves are an example of waves that involve a combination of both longitudinal and transverse motions. As a wave travels through the waver, the particles travel in clockwise circles. The radius of the circles decreases as the depth into the water increases.

Lots of wave phenomenona exhibit both longitudinal and transverse components simultaneously. For example,  

a. Seismic waves have both longitudinal and transverse components, the P and S waves. Any physical medium can support both types of mechanical wave (see reference 6 for both longitudinal and transverse waves in a piezoelectric material).

b. Ocean surface waves have both transverse and longitudinal components, and particles near the surface of the ocean execute a retrograde elliptical motion as a result (similar to the Rayleigh wave in seismology).

c. In 2002, the discovery of a trapped boundary seismic wave with both longitudinal and transverse components that travels along the ocean bottom was announced (see reference 7).

What about light and de Broglie waves?

The electromagnetic energy (light) we are familiar with is composed of transverse waves (reference 1). For example, light can be polarized, which is a feature of transverse waves. The spin states of the photon viewed as a particle are analagous to the polarization of an electromagnetic wave. However, the polarization of a photon is circular (either left or right), which is a more accurate way to visualize photon spin states (see reference 8, 9).

Normally, the longitudinal component of photons corresponds to a nonphysical gauge (see reference 2). In standard theory, as long as the photons travel at the speed of light, they cannot have a longitudinal component. However, virtual photons have a nonzero-rest mass, and travel at less than the speed of light, so they can have a longitudinal component (references 3, 4).

A Russian journal article that discusses the possibility of longitudinal electromagnetic waves in any medium can be found in reference 12.

The de Broglie waves of matter that are the basis of quantum mechanics also have spin states, something like a transverse wave (as shown in the Stern-Gerlach experiment; see reference 8). The Schrodinger equation for the propagation of these waves is a scalar equation for the wave function (whose modulus squared gives the probability of finding a particle at a given position in time and space, for example), so it does not give much indication of the nature of the wave itself. However, the more general relativistic Dirac equation is a vector PDE (or more properly, a spinor PDE), and the wavefunction in this case is a 4-dimensional spinor. Plane-wave solutions of the Dirac equation admit both positive and negative energy eigenvalues, corresponding to 2 different spin states.

In some conditions, such as having discrete time, it is possible to find longitudinal solutions called "oscillons" to the Dirac equation (see reference 10). Dirac's interpretation of the lack of time-like longitudinal solutions to his equation was connected to the existence of the time-energy uncertainty relation. As pointed out in reference 11, there are space-like longitudinal solutions to the Dirac equations.

Nikola Tesla, who appears to have headed off into pseudoscience the last few years of his career, believed that photons in a vacuum travel as a longitudinal wave.Nikola Tesla wrote: " I showed that the universal medium is a gaseous body in which only longitudinal pulses can be propagated, involving alternating compressions and expansions similar to those produced by sound waves in the air. Thus, a wireless transmitter does not propagate Hertz waves, which are a myth, but sound waves in the ether, behaving in every respect like those in the air, except that, owing to the great elastic force and extremely small density of the medium, their speed is that of light."

(Source: "Pioneer Radio Engineer Gives Views on Power", published in the New York Herald Tribune, Sept. 11, 1932, [2, p.94 ]. )

In summary, I would say that longitudinal electromagnetic fields and longitudinal solutions to the Dirac equation governing de Broglie waves are still somewhat speculative, but the interested reader can find some people pursuing this line of investigation.

When investigating this question on the internet, one has to be extremely cautious because there is an immense volume of pseudoscientific information. Probably the majority of the material on the internet on this subject is not real science.