Speed of light $> c$? lets say we have two person A and B. Person A is holding a light source (lets call this light source 'S1') which is pointed towards north. Now person A starts moving at speed 0.86c WRT (with respect to) Person B in the direction of north. Now Person A turn on the S1. what will be the speed of light emitted from S1:
1- WRT Person A.
2- WRT Person B.
This question can also be asked as an excited electron traveling at 0.86c emits a photon in the same direction in which it is moving. what will be the speed of that photon WRT to the electron and WRT to the observer (here the observer is the one from whom's respect electron is moving at 0.86c).
Please consider this hypothetical experiment that may help under the question better.
I have a metal cylinder which is 1m long. it absorbs all the light. now on its both ends i have light sensors which gives the signal to a very accurate micro-controller that can record the signal and detect the time between two events. An event is described as light being detected by a sensor. I then place this cylinder in my home horizontally and put a light source on one end. light first hit one sensor and after some time (i guess) the other sensor. My micro controller records this time as 1/299792458 seconds accurately. now i put the same experiment on a train that is moving at 0.86c. the cylinder is now placed in the direction of the train. lets say its end 1 face the engine and end 2 faces the tail and i put the light source at end 2. now i run the experiment again. now what time will be recorded by my micro controller. In this case i think it should be greater then 1/299792458 seconds. In both cases, the one at my home and the one in the train uses the same clocks and have same distance between the two sensors so this dictates that speed of light should remain same (299792458 m/s). But as both sensors are moving at speed near the speed of light, after hitting one the other is farther apart (i think due to doppler effect) the speed would change.
what will be actual time i will find in this experiment and can i use it to find my speed relative to the observer standing on the platform?
PS the above apparatus is in a vacuum chamber.
 A: Well, a photon in the vacuum always travels with the same speed c. But if you want to check, it's possible to use the relativistic velocity addition formula:
$$
v_A=\frac{u+v_B}{1+\frac{uv_B}{c^2}}
$$
where $v_S$ is the velocity measured by observer S and $u$ is the velocity of B relative to A. Now, the movement of the light source doesn't interfere with the speed at which each photon travels, so $v_B=c$.
Plugging this in, you'll find that $v_A$ is consistently also $c$.
A: Your question is ill-posed. You are obviously talking about a one-way speed, i.e. a light signal is emitted at a point $M$ at time $t_M$ and received at a point $N$ at time $t_N$ and then the speed is $v=\frac{MN}{t_N-t_M}$. But then this formula is obviously meaningless until you specify how the clocks at $M$ and at $N$ have been synchronised. Otherwise $t_N - t_M$ could take any value.
In special relativity, Einstein's procedure is the golden standard. $M$ emit a light signal at $t_M$, which is reflected by $N$ toward $M$, and $M$ gets the signal back at time $t'_M$. Then $N$ sets its clock so that the time of arrival of the light signal was $t_N=\frac{t'_M - t_M}{2}$. Now since $c=\frac{2 MN}{t'_M-t_M}$, the round-trip speed of light, is a constant independent of how $M$ and $N$ are placed, as proven to amazing precision by one century of experimental endeavours, you can see that the one-way speed of light $v=\frac{MN}{t_N-t_M}$ is also equal to $c$. 
But the reasons are completely different: the round-trip of light is a constant because of a law of nature whereas the one-way speed of light is a constant because of the convention we used to synchronise clocks.
