It was pure coincidence that we met our neighbour whilst on holiday. "In complete confidence" would imply that you are telling someone something that is confidential and is not to be repeated. (e.g. I am telling you this secret in complete confidence) "With complete confidence" would imply that you are confident in what you are proclaiming. (e.g. it is with complete confidence that I am recommending Joe for this job).

Pure means undiluted. In both cases (coincidence and confidence), 'pure' and 'complete' would amount to the same thing.

coincidence or confidence?