Solution

For @arrrchmage, @udow, @vjap55, @iamchessguy, @therealflaws and all the others who tried to solve my puzzle I will present a detailed solution now:

The first move is

1. ... Qd3!

Let me consider three main variations whereby the first one shows the actual idea of the problem.
(Of course black can checkmate immediately with 1. ... Rh1# if white plays 1. Nxd3.)

a)

2. Nxa1 (With the idea to give a deadly check at b8) 2. ... Qh3+! (The same happens after moves like for example 2. Qd4.)

3. Kxh3 Rh1# Checkmate. (Also after 3. gxh3 Rh1# wins, becaause the rook is protected by the bishop at c6.)

b)

2. Rxh5 (The idea of this move is to be able to escape via g4.) 2. ... Rh1+

3. Kxh1 Qf1+

4. Kh2 Qxg2# Checkmate.

c)

2. Rd5 (This move tries to interrupt the scope of action of the bishop c6.) 2. ... Qf1 (2. ... Qf1 also wins after moves like 2. Qc3.)

3. Rxd1 Qxg2# Checkmate. (Also after 3. Rd8+ Rxd8 4. f3 Rd1 checkmate is unavoidable.)