Does photon behave both wave like and particle like at the same time? As we all know, photons show dual character, but do they behave both ways at the same time? Then how do they propagate just like other particles do and combined with wave motion?
 A: Well, according to Heisenberg in his essay named "The physical principles of the quantum theory", the answer is more complicated that it looks. He wrote, and I agree, that our language is conditioned by our macroscopic experience. In this scale the concepts of wave and particle are perfectly defined. In the quantum world, that is, with energy on the order of the $\hbar$ (the Planck's constant) or the respective longitudinal dimensions, "waves" and "particles" start to loose their meanings, the same with "trajectory" (a concept, by the way, fundamental in the definition of particle). The answer, once we understand this, is: there exist phenomena that in certain conditions seem like waves or particles: We can't just ensure that, in fact, the physical quantum objects are waves or particles.
A: It is instructive to look at the experimental results of single photons scattering off double slits.


Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.

On the right pannel the experiment reproduces the continuous interference pattern of classical light of frequency ν. The first one on the left shows the individual photon footprints left on the camera , a dot within errors.
Each dot on the left  is the energetic footprint of a single photon, which has a "point" , i.e. particle, nature within errors of dxdydz. Nevertheless this same photon contributes to the interference pattern from the accumulation of many dots.
This tells us that on the right we see a probability density distribution, we can extract the probability of the impinging photon to be found in dxdydz of the camera. And this probability shows a wave behavior.
This is what quantum mechanics is about. Quantum mechanical entities have a location in space within errors of dxdydz, but appear at the (x,y,z) spot with a probability density given by the complex conjugate squared of the quantum mechanical solution for the specific problem,with the specific boundary conditions; in this case "photon scattering from specific double slits" .
So in this sense, photons behave like particles with a wave like probability density of materializing in space time.
Entering into  more details of the example, the footprint itself follows the solution of a secondary quantum mechanical problem, (photon scattering off camera plane molecules). This  connects the error in the width of the dots, once the detector has measuring ability in the nanometers, to the quantum mechanical heisenberg uncertainty, but this  is a nested story.
A: Well... yes and no. Photons behave like photons all the time. But when you observe a photon, you're either observing its wave-like properties or its particle-like properties, but not both at the same time. It goes back to the Heisenberg uncertainty principle, which tells us that we can either precisely observe a photon's momentum (a wave-like property) or its position (a particle-like property), but not both at the same time. Likewise, we can either precisely observe its energy (wave-like) or the timing of the photon's interaction with other particles (particle-like), but not both at the same time.
A: Don't try to visualise it as something physical. Photons are packets of light which are meant to explain the particle nature of light, and light being an electromagnetic wave explains the wave nature. 
To elaborate on this, certain phenomenon like diffraction can only be explained by consider light as an electromagnetic wave.
Other phenomenons like photo dielectric effect fails if we try to explain it using light as a wave and hence we explain using light particles or photons.
Since we know that these phenomenons occur and one nature isn't enough to explain both phenomenons, light is said to be of dual nature. 
As far as propagation of light is concerned, this phenomenon is best explained by wave nature and hence just imagine an electromagnetic wave Anne no particles
A: I think the point is at same time, since it's very hard to say what means this. We must precisely determine one moment which not exist at all. To say at same time what accident and how behaves a certain event, is a big challenge and I really sure we haven't anything named at same time!
