Can light beams (e.g. from a laser) visibly interfere if they cross in mid-air? I am wondering if two or more specific (in terms of frequency, wavelength etc) light sources could exist and capable of transmitting visible or invisible 'light', and where the beams intersect each other, can have a visible (to the naked eye) indication of interference. 
Obviously, this doesn't work with -for example- regular laser light. So I am looking if it is possible at all and if so, what kind of light is required.
Thank you.
*by mid-air I assume regular air as in our biosphere.
Very simply illustrated:
 A: Short answer: Probably not, with minor uncertainty emanating from the vagueness of your term "mid air". 
Complete answer: I would make three points:


*

*In a vacuum, the answer would be "definitely no". If you solve    Maaxwell's equations for the propagation of electromagnetic waves,    the fields corresponding to each wave just superimpose linearly; there is no way for one wave to influence the path or intensity of the other.

*Same thing in a medium where the refractive index is independent of the intensity of the light passing through it. In practically all realistic circumstances, air is such a linear medium.

*But when light that passes through a medium gets extremely intense, the response to its electromagnetic field by the electrons and nuclei of the atoms of the medium can become nonlinear, resulting in a refractive index that depends on the intensity of the field. In this case, one light beam can change the refractive index "seen" by the other, resulting in a change of path. This phenomenon is called cross-phase modulation (XPM).  I'm sorry I can't be more specific, but the gory details involved require a background in a field called nonlinear optics, which I'm unprepared to give here. 
What would it take to  accomplish this? As I said, extremely intense laser light, compressed into femtosecond pulses, might conceivably fit the bill in a sufficiently nonlinear medium. But air is almost certainly not that medium, so in practice, it's almost certainly not going to happen. Which brings us back to my short answer. 
A: As I understand the question, you would like to know if interference fringes or other indications of interference can occur **in air* due to the intersection of two light beams.  The answer is "yes".
In the link you provided, two beams intersect to provide a high-intensity region where the light power density is high enough to ionize air.  This is not interference in the sense meant by optical physicists.  However, under certain conditions a very similar experiment will produce interference.  Specifically:


*

*the beams must be mutually coherent

*the beams must be pulsed, in pulses that are brief enough that no significant changes occur due to heating of the air.

*the ionization threshold must be rather sharp (a small change in power density must produce a large change in ionization)

*the ionization lifetime - that is, the delay between absorption of light and emission of light - must be short compared to the distance air molecules (heated by the light beams) travel during the pulse duration.


If all those conditions are met, then the power of the two beams can be tuned so that ionization occurs where the beam amplitudes add (where the phases are the same), but not where the beam amplitudes subtract (where the phases are 180 degrees different).  In that case, photoemission will occur only in the regions where ionization occurs, which will be in the planes of constructive interference.
Moreover, if the beams' powers are adjusted so that simply doubling the power of the incident beam cannot cause  ionization and photoemission, but quadrupling the power will cause ionization and photoemission, then the presence of bright photoemission will be strong evidence that ionization is due to coherent interference of the two beams. That is because the power density at any point is proportional to the square of the summed amplitudes of the beams at that point.
Note that ionization is not necessary:  any nonlinearly intensity-dependent photo-absorption process will produce "mid-air" interference fringes which can in turn produce visible effects.
A: For "ordinary" lasers (visible light, milliwatt power) they will pass through one another with no fringing or other interactions. 
For powerful lasers that have wavelengths where the air is heated by their passage, then they will interact indirectly, by heating up the air along their paths and at their point of intersection, thereby changing its refractive index. 
At super-high energy levels, there are ways for the photons to interact more directly, as pointed out by Safesphere above. 
A: The word "interference" is often used for the patterns that result from waves adding up or cancelling each other when they are in or out of phase. Other answers to this question have mostly concentrated on something else: the possibility of one light wave changing the direction of another, which would only happen with very intense beams in air, and crazy-intense beams in vacuum. However, ordinary wave interference happens whenever two light waves overlap, so yes, this kind of interference does happen when two laser beams intersect. The only thing is, you won't detect it with human eyesight because the interference pattern is moving about too quickly, and the scale of the pattern is close to the wavelength of the light so our eyes cannot detect it without the aid of a microscope.
To get the interference pattern to stop moving about so quickly you need light sources with very well-defined frequency (or narrow bandwidth, as we say). The interference pattern moves through a wavelength in a time roughly given by the inverse of the frequency-spread of the light. For a typical laser the bandwidth is many megahertz so this time is sub-microsecond. In this case the interference is washed out on a timescale much faster than the human eye can respond, so it is not seen. In a high-precision optics and atomic physics lab you can get lasers with bandwidths of order hertz or less. Using these the interference pattern can be made sufficiently stable for the human eye to detect it with the aid of a microscope. 
