Necessary conditions for a given complex rational approximant R= p/q, degp, degq, to be a local best uniform approximation of a continuous complex-valued functionf defined on a compact subset of the plane are obtained. These conditions are used to characterize when a givenR is a best uniform complex rational approximant off in the special case where the extremal set off-R contains exactly max {n+degp,m+degq}+2 points.