Yes, travel insurance is a sensible default for a trip to Bishkek, and especially so if trekking into the Ala-Too range or wider Tien Shan mountains is part of the plan. Kyrgyzstan's capital sits at the foot of genuinely serious high-altitude terrain, with Ala Archa National Park barely half an hour from the city center, and that proximity is exactly why a routine capital-city stay can quickly become a policy that needs proper mountain and evacuation cover. Even a business trip confined to the city benefits from standard medical cover given how thin healthcare resources can be outside Bishkek's main hospitals.
Bishkek itself has the country's more developed hospital facilities, but care standards vary considerably once you leave the capital, and complex or specialist cases are often referred onward, sometimes to Almaty in neighboring Kazakhstan, a few hours away by road, or further abroad depending on severity. Confirm your policy explicitly covers cross-border medical evacuation and repatriation rather than assuming a routine clinic-visit benefit extends that far, since the referral chain from a mountain incident can involve more than one country.
Trekking and mountaineering in the Ala-Too range and the wider Tien Shan system are genuine draws for visitors passing through Bishkek, and altitude, remote terrain, and limited phone signal make rescue slower and costlier than a city injury would be. Check specifically for high-altitude trekking and mountain-rescue riders before assuming a baseline policy covers routes above a certain elevation, since many standard policies cap cover well below what serious Tien Shan trekking involves.
Keep any clinic paperwork, guide or permit documentation, and receipts for treatment or rescue if something goes wrong in the mountains, since claims involving remote trekking need more supporting documentation than a routine city illness. Check whether your existing health cover extends abroad and specifically to Kyrgyzstan before relying on it, and confirm your policy's evacuation benefit explicitly covers transfer across the border into Kazakhstan or onward, since that is the leg most likely to matter here.