Cause of Magnetic field According to maxwell-ampere equation, the curl of magnetic field at that point equals the current density at that point + change of electric field at that point w.r.t. time. Now, the second factor i.e. change of electric field is also caused by some moving charge at some other point.
$$
\nabla \times \vec B = \mu \vec J + \epsilon \mu  \frac {\partial \vec E}{\partial t}
$$
So is it correct to say that magnetic field are ultimately caused by currents ? ( and when the current isn't at the point where you calculated the current, you take the changing electric field factor due to that current) or can electric field be changed without reference to any charges and produce a magnetic field on its own.
 A: 
So is it correct to say that magnetic field are ultimately caused by currents ?

No, think of a magnetic field as a field that permeates all of space and time, existing independent of anything else. (However an empty field does not do anything so changes in $\vec B$ field are what matter.)
The (change in) magnetic field can be created by currents but also by other stuff too.
Using the Gauss's Law:
$$
\nabla \cdot \vec B = 0
$$
We can see that the magnetic field does not have any divergences i.e. no sources or sinks (mono-poles). You can imagine it as an incompressible fluid (like a tank of water, the water can move but you can not create high density water or vacuum volumes).
So we have established there are no divergences however there can be curls in the field, i.e. the field can flow, but since there are no divergences these flows must be closed loops.
The equation you describe, Ampère's circuital law (with Maxwell's correction), is given by:
$$
\nabla \times \vec B = \mu \vec J + \epsilon \mu  \frac {\partial \vec E}{\partial t}
$$
This states that there are two ways to create a curl in the $\vec B$ field.
The first is with a current density i.e. movement of charge which necessarily requires electric charges.
So yes (curls in) magnetic fields can be created by electric charges.
However there is a second part which states that curls in magnetic fields can also be created by $\vec E$ fields changing in time. This usually requires a charge particle but it can also be caused by a changing magnetic field. This is how light propagates, changing $\vec B$ creates a changing $\vec E$ field which creates a changing $\vec B$ etc. 
If magnetic mono-poles exist then they can be used to create the first changing $\vec E$ field. Which eliminates the need for charges (well they are magnetic charges).
A: Maxwell equations are classical electromagnetism. From a classical (not quantum) point of view, magnetism may be thought of as a relativistic consequence of a relative movement between charged particles and the frame of reference. You may want to check the Wikipedia article about it.
On the other hand, there are other sources of magnetic field which do not involve electric currents. Elemental particles have an intrinsic property, the spin, which among other things is related to it magnetic dipole moment. This property is as fundamental as the charge itself, and it can be shown that such magnetic dipole can't be accounted for by assuming any kind of spinning charge distribution. For instance, any permanent magnet work like they do because the spins of its electrons tend to align all in the same direction!
