$('#contact_id').select2({
    ajax: {
        url: 'example.com/contacts/select',
        dataType: 'json',
        delay: 250,
        data: function (params) {
            return {
                q: params.term,
                page: params.page
            };
        },
        processResults: function (data) {
            return {
                results: data
            };
        },
        cache: true
    },
    minimumInputLength: 3,
    maximumSelectionLength: 1
});

foreach($contacts as $con) {
    $results[] = [
        'id'    => $con->contact_id,
        'text'  => $con->full_name,
        'org'   => [
            'org_id'        => $con->organization_id,
            'org_name'      => $con->org_name
        ]
    ];
}

return response()->json($results);

<select>
    <option value="43" data-org="{org_id:377, org_name:'Galactic Empire'}">Darth Vader</option>
</select>

processResults: function (data) {
    data = data.map(function (item) {
        return {
            id: item.id_field,
            text: item.text_field,
            otherfield: item.otherfield
        };
    });
    return { results: data };
}

var data=$('#contact_id').select2('data')[0];
console.log(data.otherfield);

$('#select2-box').select2({
    placeholder: 'Search Existing Contacts',
    ajax: {
        url: '/contacts/typeahead',
        dataType: 'json',
        delay: 250,
        data: function(params){
            return {
                q: params.term,
                type: '',
                suggestions: 1
            };
        },
        processResults: function(data, params){
            //Send the data back
            return {
                results: data
            };
        }
    },
    minimumInputLength: 2
}).on('select2:select', function(event) {
    // This is how I got ahold of the data
    var contact = event.params.data;

    // contact.suggestions ...
    // contact.organization_id ...
});



// Data I was returning
[
    {
        "id":36167, // ID USED IN SELECT2
        "avatar":null,
        "organization_id":28037,
        "text":"John Cena - WWE", // TEXT SHOWN IN SELECT2
        "suggestions":[
            {
                "id":28037,
                "text":"WWE",
                "avatar":null
            },
            {
                "id":21509,
                "text":"Kurt Angle",
                "avatar":null
            },
            {
                "id":126,
                "text":"Mark Calaway",
                "avatar":null
            },
            {
                "id":129,
                "text":"Ricky Steamboat",
                "avatar":null
            },
            {
                "id":131,
                "text":"Brock Lesnar",
                "avatar":null
            }
        ]
    }
]