We propose a new complex block floating-point format to reduce implementation complexity. The new format achieves wordlength reduction by sharing an exponent across the block of samples, and uses box encoding for the shared exponent to reduce quantization error. Arithmetic operations are performed on blocks of samples at time, which can also reduce implementation complexity. For a case study of a baseband quadrature amplitude modulation (QAM) transmitter and receiver, we quantify the tradeoffs in signal quality vs. implementation complexity using the new approach to represent IQ samples. Signal quality is measured using error vector magnitude (EVM) in the receiver, and implementation complexity is measured in terms of arithmetic complexity as well as memory allocation and memory input/output rates. The primary contributions of this paper are (1) a complex block floating-point format with box encoding of the shared exponent to reduce quantization error, (2) arithmetic operations using the new complex block floating-point format, and (3) a QAM transceiver case study to quantify signal quality vs. implementation complexity tradeoffs using the new format and arithmetic operations.