Windows 8: Add German umlauts to US keyboard layout (maybe using AutoHotKey)

I see that the other answer was already selected, but here is my personal solution. I created a number of hotstrings. e.g. "a will give ä

:?C*:``a::à
:?C*:``i::ì
:?C*:``e::è
:?C*:``o::ò
:?C*:``u::ù
:?C*:``A::À
:?C*:``I::Ì
:?C*:``E::È
:?C*:``O::Ò
:?C*:``U::Ù

:?C*:^ :: ; Turn ^{Space} into neutral ^, else ^ will be used in next vowel.
    Send, {^}{Space}{BackSpace}
Return

:?C*:^a::â
:?C*:^i::î
:?C*:^e::ê
:?C*:^o::ô
:?C*:^u::û
:?C*:^A::Â
:?C*:^I::Î
:?C*:^E::Ê
:?C*:^O::Ô
:?C*:^U::Û


:?C*:`" :: ; Turn "{Space} into neutral ", else " will be used in next vowel.
    Send, +{'}{Space}{BackSpace}
Return

:?C*:`"a::ä
:?C*:`"i::ï
:?C*:`"e::ë
:?C*:`"o::ö
:?C*:`"u::ü
:?C*:`"A::Ä
:?C*:`"I::Ï
:?C*:`"E::Ë
:?C*:`"O::Ö
:?C*:`"U::Ü

:?C*:' :: ; Turn '{Space} into neutral ', else ' will be used in next vowel.
    Send, {'}{Space}{BackSpace}
Return

:?C*:`'a::á
:?C*:`'i::í
:?C*:`'e::é
:?C*:`'o::ó
:?C*:`'u::ú
:?C*:`'A::Á
:?C*:`'I::Í
:?C*:`'E::É
:?C*:`'O::Ó
:?C*:`'U::Ú

:?C*:`'c::ç
:?C*:`'C::Ç
:?C*:ss\::ß
:?C*:ae\::æ
:?C*:AE\::Æ
:?C*:oe\::œ
:?C*:OE\::Œ

Inspired by the last post, I figured out an additional solution, as the above sometimes caused problems while writing.

1. I first remapped CapsLock to RShift as I always use the LShift key for capital anyway
2. Setup the mapping

Rshift & a::
{
  GetKeyState, state, Lshift
  if state = D
    sendinput, Ä
  else
    sendinput, ä    
  return
}

Rshift & o::
{
  GetKeyState, state, Lshift
  if state = D
    sendinput, Ö
  else
    sendinput, ö    
  return
}

Rshift & u::
{
  GetKeyState, state, Lshift
  if state = D
    sendinput, Ü
  else
    sendinput, ü
  return
}

Rshift & s::
{
  sendinput, ß
  return
}

Rshift & e::
{
  sendinput, €
  return
}