  New hyperfields, that is fields in which addition is multivalued, are
introduced and studied. In a separate paper these hyperfields are shown to
provide a base for the tropical geometry. The main hyperfields considered here
are classical number sets, such as the set of complex numbers, the set of real
numbers, and the set of real non-negative numbers, with the usual
multiplications, but new, multivalued additions. The new hyperfields are
related with the classical fields and each other by dequantisations. For
example, the new complex tropical field is a dequantization of the field of
complex numbers.
