Adding HDMI To A USB Type-C Hub Design
Not sure if HDMI over Type-C exists, but the DisplayPort over Type-C can coexist with USB 3.1 functionality. And DP can be converted into HDMI if needed.
Type-C cables have four high-speed twisted-pair transmission channels. As a plain DP port over Type-C, all four lanes are used for full-scaled DP. In so-called &quot;Alt-DP mode&quot; two lanes are used as Rx and Tx for USB 3.x connection, and other two lanes are used for DP:

This Texas Instrument article provides more details on the variants of connectivity, and offers high-speed multiplexors to implement this.
I am not sure if there are USB3 hub controller ICs that support DP pass-through, but you certainly can use stand-alone demux-mux to make your own bypass for DP lines, provided that you re-create all necessary CC and PD control on both sides of the hub.