Elasticsearch how to use multi_match with wildcard

Alternatively you could use a query_string query with wildcards.

"query": {
    "query_string": {
        "query": "*mar*",
        "fields": ["user.name", "user.surname"]
    }
}

This will be slower than using an nGram filter at index-time (see my other answer), but if you are looking for a quick and dirty solution…

Also I am not sure about your mapping, but if you are using user.name instead of name your mapping needs to look like this:

"your_type_name_here": {
    "properties": {
        "user": {
            "type": "object",
            "properties": {
                "name": {
                    "type": "string"
                },
                "surname": {
                    "type": "string"
                }
            }
        }
    }
}