Disable "helpful" iMac message icon when text with my iPhone

So, now that I have my new iMac 5K, I'm pleased that it can text like my iPhone. My Windows PC of course cannot. BUT I do NOT want anyone sitting at my iMac to read my text messages, or even know to whom I am texting.

My question is, can I disable the nonhelpful message status icon that iMac throws up when I text on my iPhone, but still use the iMac as a text machine whenever I please?

Thanks, Ed