Robust covariance estimation

I am trying to get to grips with robust statistics. I am fitting a Gaussian to various data sets and finding that normal std is not robust enough to outliers.

For 1D I found MAD (with median: mad(x,1) in matlab) works well - and I am looking now for a multivariate equivalent.

I found FastMCD and DetMCD but these seem to be much more complicated algorithms with a lot of resampling. This is a performance critical part of the code so I think this is going to be a problem.

So at the moment I am thinking to use the MAD extended to covariance through this equation from Brown et al. which I think I can implement efficiently.

I don't have time to really look into this so wondered if someone with more experience could provide advice. Presumably the complicated MCD algorithms provide some major advantage over the multivariate MAD approach - what would they be? Is there some big flaw with the MAD method? (In 1D it looks to work quite well).

Also presumably with this approach my obtained covariance matrix will no longer be Wishart distributed so downstream considerations based on that might not apply?

Any advice appreciated.

[EDIT]: bonus figure to show the sort of data I am working with. Labelled a bit incorrectly MAD Abs = mad(x,0) in Matlab (mean deviation from the mean), MAD Med is median from median. I am multiplying mad by 1.253/1.4826 respectively to get std.