Elasticsearch.Net NEST connection string when Active Directory turn on (Elasticsearch Authentication)

It looks like you're just using Basic Authentication and sending the username/password as part of the URI. Have you tried

using Elasticsearch.Net;
using Nest;

var nodes = new [] {  
    new Uri("http://1.2.3.4:9200/")  
};

var connectionPool = new StaticConnectionPool(nodes);
var connectionSettings = new ConnectionSettings(connectionPool)
    .BasicAuthentication("username", "password");

var elasticClient = new ElasticClient(connectionSettings);

Since you only have a single node, you can also use SingleNodeConnectionPool

var connectionPool = new SingleNodeConnectionPool(new Uri("http://1.2.3.4:9200/"));