As Korpel answered already, currently there is no actual inheritance supported for Enums. So it is not possible to have a certain Enum extend and inherit the cases of another enum. 
However, I would add for completion, that Enums do support protocols, and together with protocol extensions introduced in Swift 2 and the new protocol-oriented programming approach (see this video), it is possible to implement something that resembles inheritance. This is a technique I use a lot to define UITableViewController:s driven by enums, to specify the sections of the table, and the rows within each section, and to add some useful behaviour. See for example the following sample code:
import UIKit
protocol TableSection {
    static var rows: [Self] { get }
    var title: String { get }
    var mandatoryField: Bool { get }
}
extension TableSection {
    var mandatoryTitle: String {
        if mandatoryField {
            return "\(title)*"
        } else {
            return title
        }
    }
}
enum RegisterTableSection: Int, TableSection {
    case Username
    case Birthdate
    case Password
    case RepeatPassword
    static var rows: [RegisterTableSection] {
        return [.Username, .Password, .RepeatPassword]
    }
    var title: String {
        switch self {
        case .Username:
            return "Username"
        case .Birthdate:
            return "Date of birth"
        case .Password:
            return "Password"
        case .RepeatPassword:
            return "Repeat password"
        }
    }
    var mandatoryField: Bool {
        switch self {
        case .Username:
            return true
        case .Birthdate:
            return false
        case .Password:
            return true
        case .RepeatPassword:
            return true
        }
    }
}
class ViewController: UITableViewController {
    override func viewDidLoad() {
        super.viewDidLoad()
    }
    override func numberOfSectionsInTableView(tableView: UITableView) -> Int {
        return 1
    }
    override func tableView(tableView: UITableView, numberOfRowsInSection section: Int) -> Int {
        return RegisterTableSection.rows.count
    }
    override func tableView(tableView: UITableView, cellForRowAtIndexPath indexPath: NSIndexPath) -> UITableViewCell {
        guard let row = RegisterTableSection(rawValue: indexPath.row) else {
            // This should never happen
            return UITableViewCell()
        }
        let cell = UITableViewCell()
        cell.textLabel?.text = row.mandatoryTitle
        return cell
    }
}
The previous code would render the following table:

Notice how by implementing the protocol, our RegisterTableSection enum has to provide implementations to methods and variables defined in the protocol. And most interestingly, it inherits a default implementation of the variable mandatoryTitle through the TableSection protocol extension
I have uploaded the source code of this example here