Is there a hash function with 2048bit output? Is there a publicly available cryptographic hashing algorithm with 2048 bit output?

The standard ones are "only" up to 512 bit (SHA-512, WHIRLPOOL).

(2048 bits are 256 bytes, so it would be useful for generating a full-length key for the ARC4 cipher)
I was already wondering why you would need 2048 bits, because brute forcing a 512 bit output hash takes already a $2^{256}$ operations on average (w.r.t to collision resistance, due to the Birthday Paradox – preimage and second-preimage attacks take on average $2^{511}$ hashes), which is secure enough, for all kinds of hardware available today.

I assumed that you would probably want to transform the hash output into some form of key with a length of 2048 bits, your comment confirmed this. In that case, what you need is not necessarily a hash output of that length, but you need a secure Key Derivation Function (KDF) that allows "stretching" the random hash output to the desired size. 

Have a look for example at the KDF described in NIST Special Publication 800-56A: “Recommendation for Pair-Wise
Key Establishment Schemes
Using Discrete Logarithm
Cryptography ” that describes how to stretch hash outputs to arbitrary sizes securely.

When equipped with such a KDF, it basically doesn't matter what hash function you are using (as long as it itself is secure) so you can safely use one of the sha-2 family hashes (SHA-256, 512 etc.).