The developer of a luxury apartment complex on the site of the former Children’s Museum in West Hartford has paid $10.6 million for the property, with construction expected to begin this spring. The price paid by New York-based Continental Properties for the nearly four-acre property at 950 Trout Brook Drive near Farmington Avenue is about $2.7 million an acre — a price, brokers involved in the sale say, sets a new benchmark for development sites in central Connecticut. In Connecticut, Continental has developed upscale rental complexes in Rocky Hill, Glastonbury, South Windsor, Milford, Shelton and Trumbull. Meanwhile, plans call for the iconic, 60-foot, 20-ton Conny the Whale sculpture to be moved just across the street to the Trout Brook greenway. Conny was constructed in the mid-1970s as a symbol of the “Save the Whales” movement but had evolved into a playscape and mascot for the museum.