I've recently made a solution where the customer requested to be able to bring up the calendar view in a DateTimePicker control by pressing on a specific button on the screen. The solution to that was really simple: Create a control that inherits from System.Windows.Forms.DateTimePicker and add a method called ShowCalendar() which I call to bring up the Calendar view.

public class DateTimePickerEx : DateTimePicker
  static extern int SendMessage(
    IntPtr hWnd, uint uMsg, int wParam, int lParam);

  const int WM_LBUTTONDOWN = 0x0201;

  public void ShowCalendar() {
    int x = Width - 10;
    int y = Height / 2;
    int lParam = x + y * 0x00010000;

    SendMessage(Handle, WM_LBUTTONDOWN, 1, lParam);