Is force frame dependent? If we look at a particle moving in the positive $x$ direction from a frame that is accelerating in the positive $y$ direction then its acceleration will be different from that in an inertial frame and hence the force on the particle must also be different in this frame. 
Is this right? 
 A: As described in the question, you have only said that the frame is moving. But if motion of the frame is uniform, then acceleration is not changed and therefore the force is not changed (Newton II).
If the frame is accelerating relative to an inertial frame, then there will be an inertial force (or pseudo force). Inertial forces are distinct from active forces (inertial literally means not acting). The inertial force accounts for the difference in the acceleration of the particle when seen from the accelerating frame.
A: All forces can be categorised under the four fundamental forces(or a combination of them). 
1.Gravitation
2.Electromagnetic
3.Stong Nuclear force
4.Weak Nuclear force
Let us consider an example - Gravitational force. 
$$F=\frac{Gm_1m_2}{r^2}$$
G is an universal constant(independent of frame of reference)
$m_1$ and $m_2$ are also frame independent. r-the separation between the 2 point masses is also independent(displacement is frame dependent, but not separation). Hence the net force will also be independent of the frame of reference. Similarly one can argue for the other forces.
Hence we consider Force to be independent of frame of reference.
A: Note: this answer is based on Newtonian mechanics not special or general relativity. 
If we start in an inertial frame and consider a particle acted on by a real force in the $x$ direction then the motion of the particle is given by 
$m \ddot x=F$
$m \ddot y=0$
$m \ddot z=0$
Now, consider a reference frame initially at rest with respect to the inertial frame and accelerating at $a$ in the $y$ direction then to transform between coordinates we can use
$X=x$
$Y= y -\frac{1}{2}a t^2$
$Z=z$
So in this frame the motion of the particle is given by
$m \ddot X =m \ddot x = F$
$m \ddot Y =m \ddot y -ma=-ma$
$m\ddot Z=m\ddot z=0$
So in the accelerated frame the same real force $F$ acts in the $X$ direction but in addition there is a term that looks like a force $-ma$ acting along $Y$ which is the direction of the non-inertial frame’s acceleration. 
This term is often called a fictitious force or an inertial force, but although it is called a force it is an artifact of the reference frame and not an interaction between objects, meaning that it does not follow Newton’s 3rd law. 
So in a non inertial reference frame the real force still exists but in addition there is an inertial force present. 
A: I will answer this with Galilean relativity for Newtonian mechanics. These are elementary boosts with $\vec x\rightarrow \vec x+\vec ut$ and velocity change $\vec v\rightarrow \vec v+\vec u$. Newtonian force is $\vec F=\frac{d\vec p}{dt}$, such that momentum is $\vec p=m\vec v$. Momentum is the frame dependent, but clearly because $\frac{d\vec u}{dt}=0$ force is not frame dependent. It also means that momentum, while frame dependent, does not increase or decrease with time due to a change of frame, and is thus conserved.
