IQ imbalance amplitude
What it looks like you are experiencing is carrier offset and not IQ imbalance, at least from this view that you show. Looking at the scale of the two axis it looks balanced and it is just that each axis is drawn at a different scale? 

See my example plot below of QPSK before and after proper carrier recovery. Note that it also helps to show the correct symbol locations (from your timing recovery which can be done under carrier offset conditions), which are indicated by the red dots in the plot.



For more details on implementations to remove the carrier offset please see my post here: 

High modulation index PSK - carrier recovery

I am interested in what particular modulation you are actually using as there may be more direct methods we could suggest, even a cross correlation of your signal with a delayed version of itself can reveal the frequency offset for example, such as depicted in the figure below), which I would be tempted to do in a post processing analysis for quickly accessing what the carrier is, that or just discerning the rate of rotation from the trajectory:



Once you successfully "stop it from spinning", here are practical techniques to remove offset errors (amp and phase imbalance) if you indeed have that problem:

First, this is what IQ imbalance would look like (in addition you could also have a "DC offset" which would appear as a stronger carrier in your spectrum):



And here is a simple technique for IQ imbalance correction:



With the techniques to establish the $\alpha$ and $\beta$ coefficients in the plots below: