Same as after Nd5, d4 looks good. Then I guess the idea for black is too attack the central pawns so d6 Nf3 to protect. If dxe5, Nxe5 to avoid the queen trade.
What would be the best move for white from here?
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3. d4 does threaten to capture the knight, but 3 ... h6 is not the best reply for black i think. A bit passive? Better is 3 ... e6 because 4. f3 can now be met by Qh5 check.
It looks like a trap opening to me. It challenges white to trap black's knight but it is harder than it looks.
I don't have an engine but if I did, I'd check whether this opening is objectively bad for black and what the engine refutation is. But I suspect that this opening is "playable" for black and trying too hard to trap the black knight may be the wrong strategy.
Thanks for all the helpful comments :)
Nd6 is another commen move in this position. Would d4 be the best reply to that? Or to both actually..?
But then who is having an advanteous position here? White moved the pawn twice, black has a night in white's teritory...
Is e4-e5 the best possible move here?
This opening is called Alekhine's defence. It's a standard opening by black, so probably no one has the advantage yet. e5 is not a bad move. The most common reply is Nd5 though, not Ne4. After Nd5, common moves for white include c4 and d4, and I believe there are others too. That's where my theory ends unfortunately.
Thanks Keju, that helped a lot!
Sure thing, this may also help if you haven't googled it already: http://en.wikipedia.org/wiki/Alekhine's_Defence
A simple opening & defense for white and black to make a mating pattern in lightning :
http://www.chess.com/livechess/game?id=652216868
http://www.chess.com/livechess/game?id=652354893
That line is called the Mokele M'bembe, a provocative move to lure white into a weakening move to supposedly trap the knight, which does not always work out so well. The safe line is d3, followed by d4.
Now, d4 with the threat of f3!? is certainly worth a look, but be warned that the complications might be unclear. If you play it wrong, white will be subjected to an attack instead. If you play it well, black may have to sac the knight for insufficient compensation. I think I have that line somewhere, but I don't think the issue is so difficult you cannot figure it out.
Thanks Dutchday, what a great name for this variation 
Yes it's an African mythological creature that may well have been a hippo, a crocodile, or indeed a mythological creature that has eluded mankind.
Great info everyone, thanks ;)
check out danny rensch's new live session for a game against a GM playing 2...Nd5. he played two games though instead of giving an overview of the theory afterwards, but pretty interesting
1.e4 Nf6 2.e5 Ne4 3.d3 Nc5 4.d4 looks fine. (4...Ne4 5.f3) Maybe early f3 has some really complicated point to it but seeing as 1...Nf6 is rare and 2...Ne4 is even rarer I'd prefer a simple approach with no drawbacks.
1.e4 Nf6 2.e5 Ng8 3.d4 probably leads to normal positions with an extra tempo.
I think Ng8 is better than Ne4 since the extra tempo might not be so bad. With Ne4 your knight is all over the place and you have to play really weird moves to keep it safe.
A simple e4 opening for white. What would be the best next few moves for white? I recently got into this position a few times and blundered...
Thanks for having a look.