Anderssen's 'immortal game' I came across Anderssen's 'immortal game':

[FEN ""]
1. e4 e5 2. f4 exf4 3. Bc4 Qh4+ 4. Kf1 b5
5. Bxb5 Nf6 6. Nf3 Qh6 7. d3 Nh5 8. Nh4 Qg5
9. Nf5 c6 10. g4 Nf6 11. Rg1 cxb5 12. h4 Qg6
13. h5 Qg5 14. Qf3 Ng8 15. Bxf4 Qf6 16. Nc3 Bc5
17. Nd5 Qxb2 18. Bd6?


I was curious to know whether Bd6 is really a good move or not. StockFish 10+ does not think it is all that great. At the default depth 18 it evaluates it wrongly (strangely it gave different values whenever I forced reevaluation, but typically between -4 and -5), but even at depth 29 it still evaluates it as 0, with main line 18... Qxa1+ 19. Ke2 Qb2 20. Bxc5 Qxc2+ 21. Kf1 Qxc5 22. Qf4 Kd8 23. Nd6 f6 24. g5.

But it seems even 18... Qxa1+ 19. Ke2 Bxg1 20. e5 is not too bad for Black, and I think continues:

[FEN ""]
1. e4 e5 2. f4 exf4 3. Bc4 Qh4+ 4. Kf1 b5
5. Bxb5 Nf6 6. Nf3 Qh6 7. d3 Nh5 8. Nh4 Qg5
9. Nf5 c6 10. g4 Nf6 11. Rg1 cxb5 12. h4 Qg6
13. h5 Qg5 14. Qf3 Ng8 15. Bxf4 Qf6 16. Nc3 Bc5
17. Nd5 Qxb2 18. Bd6? Qxa1+ 19. Ke2 Bxg1 20. e5 Ba6
21. Nc7+ Kd8 22. Nxa6 Qc3 23. Bc7+ Qxc7 24. Nxc7 Nc6
25. Nxa8


It seems Black is not really dead yet. Am I missing some subtleties?

I also noticed that the wikipedia article has an apparently inconsistent footnote [6], which implies that 18...Qxa1+ is better because it likely leads to 19. Ke2 Bxg1 20. e5., but this is precisely the same position that Anderssen reached in the actual game, which the article says led to Black's defeat...
You are correct that 18.Bd6? is, indeed, not the best move (the simple Re1! threatens Nc7-a8 winning on material). Bd6 has widely been considered a mistake for many years now. Black holds with 18...Qxa1+ 19.Ke2 Qb2! 20.Bc5 Qc2 21.Kf1 Qc5 22.Qf4 and black has several moves that are equal, but in the last few moves, it is white that has to find the only moves along the way.

[fen "rnb1k1nr/p2p1ppp/3B4/1pbN1N1P/4P1P1/3P1Q2/PqP5/R4KR1 b - - 0 1"]

1...Qxa1+ 2.Ke2 Qb2! 3.Bc5 Qc2 4.Kf1 Qc5 5.Qf4


Of course, this was played at a time where "romantic chess" was still at its peak, and winning practically was not their mindset. It was sacrifice, and if your opponent offered pieces, you took them.

Lastly, with computers getting stronger every year, who knows, maybe 18...Qa1 will eventually still be found to be winning for white, but that is a moot point since Re1 kept the material and the attack.