Best method to compare indexes I have implemented three different estimates to measure the same character, and I would like to statistically compare them (to see if they indeed measure the exact same signal, or if they give different pieces of information).
How can I do that?
Correlation test always find a significant correlation, but since the estimates measure the same thing...
You can take a look at the dataset here
 A: Now that the data have been posted some simple analyses help make the situation clearer. 
Let's show (half of) a scatter plot matrix. 

There is a lot of scatter. A and B are clearly most similar; C is largely dissimilar to B but shows more relation to A. There aren't problems with extraordinary outliers or massive curvature. 
Concordance correlations quantify this without adding insight: 
A B  0.704 
B C  0.363 
A C  0.638   

Concordance correlations measure agreement, not linearity, and are 1 if and only if two variables have identical values. 
The principal component analysis I would base on a covariance matrix, as it seems that the units of measurement are identical. These results (from Stata) don't match those in the OP's comments: 
. pca A B C, cov

Principal components/covariance                   Number of obs    =        70
                                                  Number of comp.  =         3
                                                  Trace            =  .1942012
    Rotation: (unrotated = principal)             Rho              =    1.0000

    --------------------------------------------------------------------------
       Component |   Eigenvalue   Difference         Proportion   Cumulative
    -------------+------------------------------------------------------------
           Comp1 |      .153565      .123528             0.7908       0.7908
           Comp2 |     .0300363     .0194362             0.1547       0.9454
           Comp3 |     .0106001            .             0.0546       1.0000
    --------------------------------------------------------------------------

Principal components (eigenvectors) 

    ----------------------------------------------------------
        Variable |    Comp1     Comp2     Comp3 | Unexplained 
    -------------+------------------------------+-------------
               A |   0.6037    0.1114   -0.7894 |           0 
               B |   0.4836    0.7360    0.4738 |           0 
               C |   0.6337   -0.6678    0.3904 |           0 
    ----------------------------------------------------------

So, with a covariance matrix input, PCA shows PC1 with 79% of the total. 
(Using a correlation matrix produces the same rounded result of 79% for the first PC; from the graphs it is evident that the variances of the variables are similar.) 
I wouldn't personally mush together A, B and C and take PC 1 as the best measure of a latent variable. Either C is the poorest measure and should be discarded and A and B combined; or there might be quite independent grounds for thinking that C is the best method, in which case it should be used. Consider three witnesses in court: two liars might agree with each other and disagree with one honest person. The majority need not be correct. Conversely, on a consensus or coherence criterion C looks poorest. 
EDIT: Here is a quantile plot, which underlines what the scatter plots do show, but less clearly, which is that C is typically smallest. 

