It has been already mentioned by Victor Protsak the existence of application to coding theory (Tsfasman's & K works). But let me mention yet another application to this field which is more recent and being under development at the present.
One may see papers by F. Oggier, G. Rekaya-Ben Othman, J.-C. Belfiore, E. Viterbo: e.g. this one : http://arxiv.org/abs/cs/0604093.
Unfortunately I do not fully understand the idea but let me give brief comments.
Consider multi-antenna transmission and receiver (MIMO for short http://en.wikipedia.org/wiki/Mimo ) , this means we have vector (x1...x_{transmit antenna number} ) and received vector Y=(y_1... y_{receive antenna number} ).
The received signal is Y= H X + noise. where H - matrix of size receive antenna by transmit antenna, noise - is random noise vector.
The nature of our signal is discreete so we should choose some LATTICE in C^{transmit antenna} and actually sent signal is one the lattice points - not an arbitrary vector. Of course, it is not full lattice but truncated to some finite area due to power constraint.
So the problem is to select what kind of lattices are suitable for the best quality of transmission ? Moral of these papers that we should choose lattices of algebraic integeres in specifically selected algebraic fields like $Z(\sqrt n_1, \sqrt n_2) \subset Q(\sqrt n_1, \sqrt n_2)$.
PS
Actually I am significalntly over-simplifying the situation. We should also take into account time dimension. The the actual space is NOT C^{transmit antenna}, but C^{transmit antenna * N}, where N is block length which we choose. This is related to the so-called "space-time codes" http://en.wikipedia.org/wiki/Space%E2%80%93time_block_code
PSPS The story of space time codes starts from the so-called Alamouti code http://en.wikipedia.org/wiki/Space%E2%80%93time_block_code#Alamouti.27s_code
Looking on the matrix in Wikipedia you can immediately reconginze the quaternions presented by the 2x2 complex matrices. Let me mention that it is mandatory for implematation in 3G smartphones. So we can say that smartphones know quaternions :)
If these works will be succussful every smartphone will "know" algebraic number theory :)
